 \(x_{n}/n \rightarrow \infty \) , \(G\)  is \(\Delta \) -analytic and there exist \(\alpha , c>0\)  such that \(G(\rho -z) \sim c/z^{\alpha }\)  as \(z \rightarrow 0\)  with \(\mathrm {Re}(z)>0\) , \(v_{n} = x_{n}^{2}/(\alpha n)\) , and finally \(C=\alpha +1\) .
As already mentioned, the functional convergence in the first case is essentially contained in [1]}.
On the other hand, to the best of our knowledge, the two other cases are new.
Theorem REF  is rather intuitive: if we force the path to end at a very low value, then first \(p(0)\)  has to be nonzero, and as alluded after Theorem REF  the assumption \(p(1)>0\)  prevents some periodicity in the large scale, and finally in the limit the path must not make any large jump, hence a Brownian limit.
On the other hand, in Theorem REF , we condition the path to end at a very high value, and still we obtain a Brownian limit; this unusual behaviour comes from the assumptions on the weights (which, as noted in the introduction, naturally appear when looking at uniform random maps), and Theorem REF  below exhibits a very different behaviour.
In our last main result on these random paths, we study the boundary case \(x_{n}/n \rightarrow \Psi (\rho )\)  when the latter is finite. To do so, we assume that the probability distribution with generating function \(G(\rho \,\cdot )/G(\rho )\)  belongs to the domain of attraction of a stable law with index \(\alpha \in (1, 2]\) , i.e. that there exist \(\alpha \in (1,2]\) , \(\rho >0\) , \(m>0\) , and a slowly varying function \(L\)  such that for every \(0 \leqslant s < \rho \) ,
\(G(s)=G(\rho ) \mathopen {}\mathclose {\left(1-m+\frac{ms}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{\alpha } L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\right)}.\) 
In the case \(G^{(2)}(\rho ) < \infty \) , i.e. when \(\alpha =2\)  and \(L\)  admits a finite limit at infinity, say \(\ell \) , the law with generating function \(G(\rho \,\cdot )/G(\rho )\)  has a finite variance, equal to
\(\sigma ^{2} = 2\ell + m - m^{2}= \frac{\rho ^{2} G^{(2)}(\rho )}{G(\rho )} + \frac{\rho G^{\prime }(\rho )}{G(\rho )} - \mathopen {}\mathclose {\left(\frac{\rho G^{\prime }(\rho )}{G(\rho )}\right)}^{2}.\) 
In this case we set \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  for every \(n\geqslant 1\) , otherwise, we let \((r_{n})_{n \geqslant 1}\)  be a sequence such that \(r_{n}^{\alpha } \sim n L(r_{n})\)  as \(n \rightarrow \infty \) .
It is well-known (see e.g. [2]} and [3]}) that if \((S_{n})_{n\geqslant 1}\)  is a random walk with i.i.d. increments with generating function \(G(\rho \,\cdot )/G(\rho )\) , then the convergence
\(\mathopen {}\mathclose {\left(\frac{1}{r_{n}} \cdot \mathopen {}\mathclose {\left(S_{{nt}} - mnt\right)} ; t \geqslant 0\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }X^{\alpha },\) 
holds in distribution in \([0,\infty ), \mathbb {R})\) . See the remarks around (REF ) for the scaling.
Recall from Sec. REF  that for \(\lambda \in \mathbb {R}\) , we denote by \(X^{\alpha ,\lambda ,\mathrm {br}}\)  the process \((X^{\alpha }_{t}-\lambda t)_{t}\)  conditioned to end at 0 at time 1.
Let us also denote by \(\Delta X^{\alpha ,\lambda ,\mathrm {br}}_{t} = X^{\alpha ,\lambda ,\mathrm {br}}_{t}-X^{\alpha ,\lambda ,\mathrm {br}}_{t-}\)  the value of the `jump' of \(X^{\alpha ,\lambda ,\mathrm {br}}\)  at time \(t \in [0,1]\) , which is null except for a countable set of times.
The functional convergence in distribution in item (iii) below is already known (see the discussion just afterwards), but we include it here for completeness.
Theorem 4.3 
Under the preceding assumptions, if \(x_{n}\)  takes the form
\(x_n = m n + \lambda _{n},\) 
where
\(\lambda _{n}/r_{n}\)  converges to some \(\lambda \in [-\infty ,\infty ]\) , then three cases occur.
 If \(\lambda \in \mathbb {R}\) , then
\( \mathopen {}\mathclose {\left( \frac{1}{r_{n}} \cdot \mathopen {}\mathclose {\left(S^{n}_{{nt}} - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }X^{\alpha ,\lambda ,\mathrm {br}}.\) 
In addition,
\(r_{n}^{-2} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\begin{dcases}2 + \frac{2 m^{2}}{\sigma ^{2}} \mathbb {1}_{\lbrace G^{(2)}(\rho ) < \infty \rbrace } &\textrm { if } \alpha = 2,\\\sum _{t \in [0,1]} (\Delta X^{\alpha ,\lambda ,\mathrm {br}}_{t})^{2} &\textrm { if } \alpha <2.\end{dcases}\) 

 If \(\lambda = -\infty \)  and if \(\lambda _{n}/n \rightarrow 0\) , then with \(\varepsilon _{n}\)  such that \(\Psi (\rho (1-\varepsilon _{n})) = x_{n}/n\) , we have
\( \mathopen {}\mathclose {\left(\sqrt{\frac{1}{\alpha -1} \frac{\varepsilon _{n}}{|\lambda _{n}|}} \cdot \mathopen {}\mathclose {\left(S^{n}_{{nt}} - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }B^{\mathrm {br}}.\) 
In addition,
\(\frac{\varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\alpha -1 + \frac{m^{2}}{\sigma ^{2}} \mathbb {1}_{\lbrace G^{(2)}(\rho )<\infty \rbrace }.\) 

 If \(\lambda = \infty \) , we further assume that \(p(k)\rho ^{-k}\)  is regularly varying at infinity with some index \(-\beta < -2\) .
If \(G^{(2)}(\rho )<\infty \)  (which implies \(\beta \geqslant 3\) ), we furthermore assume that there exists \(c>(\beta -3)/\sigma ^{2}\)  such that \(\lambda _{n} \geqslant \sqrt{c \ln (n)}\) .
Then
\(\mathopen {}\mathclose {\left(\frac{1}{\lambda _{n}} \cdot \mathopen {}\mathclose {\left(S^{n}_{{nt}} - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }(\mathbb {1}_{\lbrace U \leqslant t\rbrace } - t; 0 \leqslant t \leqslant 1),\) 
where \(U\)  is uniformly distributed on \([0,1]\) .
In addition,
\( \frac{\max \lbrace X^{n}_{1},X^{n}_{2}, \ldots , X^{n}_{n}\rbrace ^{2} }{\sum _{k=1}^{n} (X^{n}_{k})^{2} } \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1.\) 

Let us make some comments. In the first regime, we condition the path \(S^{n}\)  to end at a value which only differs from its average behaviour by some constant \(\lambda \)  times \(r_{n}\) , which is the scaling of the natural fluctuations as we recalled before the statement, so the scaling remains the same and the limit is simply the same Lévy process as in the unconditioned case, but conditioned to end at \(\lambda \) . In the second regime however we condition the path to end at a much lower value (we assume \(\lambda _{n}/n \rightarrow 0\)  as otherwise we fall in the bulk regime of Theorem REF ) and this completely changes the scaling, from \(1/r_{n}\)  to \(\sqrt{\varepsilon _{n}/|\lambda _{n}|}\) , which, as we shall later see, is of order \(1/r_{n}\)  in the finite variance case, but larger otherwise;
this wipes away the natural behaviour of \(G\)  and it yields in fact a Gaussian limit at this scaling, which can be understood intuitively, as for Theorem REF , by the fact that, in order to end at a low value, the path must avoid any large jump.
Finally, in the last regime, a `one big jump principle' holds, which contrasts with Theorem REF . The functional convergence in this regime is already known (see e.g. [4]}). However we will need a more detailed analysis, which follows from a direct application of the results from [5]}, [6]}:
In addition to a unique giant increment, of size asymptotically equivalent to \(\lambda _{n}\)  (as the combination of the two convergences shows), occurring at a random time asymptotically uniformly distributed, the \(n-1\)  other increments are close (in total variation) to just i.i.d. random variables, with generating function \(G(\rho \,\cdot )/G(\rho )\) , so the path obtained by removing this giant increment stays in the scale \(r_{n}\) .
We refer to Sec. REF  below for details.
Functional convergence from a local limit estimate
Let us now prove how the Lévy bridges appear as limits of our discrete paths, based on the local limit estimates from Sec. . Since the same argument applies to prove most of them, we present a general result that we then apply to our case. We denote by \(X\)  a stable Lévy process with a linear drift, with a density \(d_{t}\)  at time \(t\) , and we let \(X^{\mathrm {br}}\)  denote a version of \(X\)  conditioned on \(X_{1} = 0\) , whose definition was given in Sec. REF .
For the proof of Theorems REF , REF , REF , and REF  we shall take \(X\)  to simply be a standard Brownian motion without drift.
Proposition 4.4 (Subcritical conditioning) 

Suppose that \(x_{n}/n < \Psi (\rho )\)  for every \(n\)  large enough and let \(b_{n} = \Psi ^{-1}(x_{n}/n)\) . Assume that there exists a sequence \(v_{n} \rightarrow \infty \)  such that for every \(t > 0\) ,
\(\sup _{k \geqslant 0} \mathopen {}\mathclose {\left|v_{n} \cdot \frac{b_{n}^{k}}{G(b_{n})^{{nt}}} \cdot \mathbb {P}\mathopen {}\mathclose {\left(S^{n}_{{nt}} = k\right)} - d_{t}\mathopen {}\mathclose {\left(\frac{k-x_{n}t}{v_{n}}\right)}\right|} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
Then the convergence
\(\mathopen {}\mathclose {\left( \frac{1}{v_{n}} \mathopen {}\mathclose {\left(S^{n}_{\lfloor n t \rfloor } - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }X^{\mathrm {br}}\) 
holds in distribution in \([0,1],\mathbb {R})\) .

By construction of \(b_{n}\) , we can define a random variable \(\widehat{X}^{n}\)  by setting for every \(k \geqslant 0\) ,
\(\mathbb {P}\big (\widehat{X}^{n} = k\big ) = \frac{1}{G(b_{n})} b_{n}^{k} p(k),\qquad \text{so}\qquad \mathbb {E}\big [\widehat{X}^{n}\big ]= \Psi (b_{n})= \frac{x_{n}}{n}.\) 
Furthermore, according to Lemma REF , the random walk \(\widehat{S}^{n} = (\widehat{S}^{n}_{i} ; 0 \leqslant i \leqslant n)\)  with step distribution \(\widehat{X}^{n}\) , when conditioned on \(\widehat{S}^{n}_{n} = x_{n}\) , has the same law as our original path \(S^{n}\) .
Now the local limit estimate (REF ) can be rewritten as
\(\sup _{k \geqslant 0} \mathopen {}\mathclose {\left|v_{n} \cdot \mathbb {P}\mathopen {}\mathclose {\left(\widehat{S}^{n}_{{nt}} = k\right)} - d_{t}\mathopen {}\mathclose {\left(\frac{k-x_{n}t}{v_{n}}\right)}\right|} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
Then this implies the convergence in distribution of \(v_{n}^{-1} (\widehat{S}^{n}_{{nt}} - x_{n} t)\)  to \(X_{t}\) , under the unconditional law, for any \(t>0\)  fixed, which, by e.g. [3]}, actually suffices to conclude to the convergence of the whole path, still under the unconditional law, namely
\(\mathopen {}\mathclose {\left(\frac{1}{v_{n}} \mathopen {}\mathclose {\left(\widehat{S}^{n}_{{nt}} - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }(X_{t} ; 0 \leqslant t \leqslant 1).\) 
We deduce the convergence of bridges on any time interval \([0,u]\)  with \(u \in (0,1)\)  by absolute continuity. Indeed, let us set \(\varphi ^{n}_{i}(k) = \mathbb {P}(\widehat{S}^{n}_{i} = k)\)  for every \(i\)  and \(k\)  and let \(F : [0,u],\mathbb {R}) \rightarrow \mathbb {R}\)  be a bounded continuous functional, then by (REF ) first and then (REF ),

E[F(1vn (Snnt - xn t) ; 0 t u) | Snn=xn]
= E[F(1vn (Snnt - xn t) ; 0 t u) nn-nu(xn -Snnu)nn(xn)]
= E[F(1vn (Snnt - xn t) ; 0 t u) d1-u(- vn-1 (Snnu - xn u))d1(0)] (1+o(1))
= E[F(Xt ; 0 t u) d1-u(-Xu)d1(0)] (1+o(1)).

Notice that \((\widehat{S}^{n}_{n}-\widehat{S}^{n}_{n-i})_{ 0 \leqslant i \leqslant n}\)  has the same distribution as \((\widehat{S}^{n}_{i})_{ 0 \leqslant i \leqslant n}\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}^{n}_{n}=x_{n})\) , so the convergence of bridges also holds on the time interval \([u,1]\) . In particular, the portion of the path on \([u,1]\)  is tight and we may extend the preceding convergence in distribution of the bridges to the whole interval \([0,1]\) .
This shows our claim when \(S^{n}\)  is replaced by the bridge of \(\widehat{S}^{n}\)  and we conclude the proof by recalling that these paths have the same law.
With this result at hand and the local estimates from Sec. , we can easily prove some of our main results.
[Proof of Theorems REF  and REF ]
Let us first prove Theorem REF ; under its assumptions, we deduce from Theorem REF  and Proposition REF , the convergence
\(\mathopen {}\mathclose {\left( \frac{1}{\sqrt{\widehat{\sigma }_{n}^{2} n}} \mathopen {}\mathclose {\left(S^{n}_{\lfloor n t \rfloor } - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }B^{\mathrm {br}},\) 
where \(\widehat{\sigma }_{n}^{2}\)  is the variance of \(\widehat{X}^{n}\)  in the above notation, which is given by (REF ); by continuity it converges to the same expression with \(b\)  instead of \(b_{n}\) , i.e. \(\sigma ^{2}_{b}\)  in the notation of Theorem REF .
Similarly, Theorems REF , REF , and REF  immediately follow by combining Proposition REF  with Theorem REF , Theorem REF , and Theorem REF  respectively.
In each case, the proof of the convergence of the sum of the increments squared is deferred to Appendix , see Proposition REF , Proposition REF , Proposition REF , and Proposition REF  respectively. In each regime, the basic idea is to control the first four moments of \(\widehat{X}^{n}\) , which are expressed in (REF ).
We next modify the preceding statement in order to control the case where \(x_{n}/n\)  is slightly larger than \(\Psi (\rho )\) , which will enable us to treat Theorem REF .
For \(c \in \mathbb {R}\)  and \(X\)  the preceding Lévy process, recall that we define \(X^{c,\mathrm {br}}\)  as the process \((X_{t} - ct ; 0 \leqslant t \leqslant 1)\)  conditioned to end at 0.
Proposition 4.5 (Critical conditioning) 

Assume that \(\Psi (\rho ) < \infty \)  and that there exists a sequence \(v_{n} \rightarrow \infty \)  such that for every \(t > 0\) ,
\(\sup _{k \geqslant 0} \mathopen {}\mathclose {\left|v_{n} \cdot \frac{\rho ^{k}}{G(\rho )^{{nt}}} \cdot \mathbb {P}\mathopen {}\mathclose {\left(S^{n}_{{nt}} = k\right)} - d_{t}\mathopen {}\mathclose {\left(\frac{k-\Psi (\rho ) n t}{v_{n}}\right)}\right|} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
If there exists \(c \in \mathbb {R}\)  such that \(v_{n}^{-1} (x_{n} - \Psi (\rho ) n) \rightarrow c\) , then the convergence
\(\mathopen {}\mathclose {\left( \frac{1}{v_{n}} \mathopen {}\mathclose {\left(S^{n}_{\lfloor n t \rfloor } - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }X^{c, \mathrm {br}}\) 
holds in distribution in \([0,1])\) .

Let us replace \(b_{n}\)  in the preceding proof by \(\rho \) , i.e. define a random walk \(\widehat{S}\)  with step distribution \(\widehat{X}\)  given by
\(\mathbb {P}\big (\widehat{X} = k\big ) = \frac{1}{G(\rho )} \rho ^{k} p_{k}\qquad \text{for every}\qquad k \geqslant 0.\) 
Note that its mean equals \(\mathbb {E}[\widehat{X}] = \Psi (\rho )\) , then the argument that lead to (REF ) applies here and we deduce, using also \(v_{n}^{-1} (x_{n} - \Psi (\rho ) n) \rightarrow c\) , that
\(\mathopen {}\mathclose {\left(\frac{1}{v_{n}} \mathopen {}\mathclose {\left(\widehat{S}_{{nt}} - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }(X_{t} - ct ; 0 \leqslant t \leqslant 1),\) 
under the unconditional law.
As previously, this then transfers to the bridge conditioning by absolute continuity and time-reversal.
The convergence of paths in Theorem REF  then immediately follows as the preceding theorems.
[Proof of Theorem REF ]
As the preceding case, we deduce immediately from (REF ) and Proposition REF  the convergence
\(\mathopen {}\mathclose {\left(\frac{1}{r_{n}} \mathopen {}\mathclose {\left(S^{n}_{\lfloor n t \rfloor } - x_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }X^{\alpha ,\lambda ,\mathrm {br}}.\) 
Let us next prove the convergence of the sum of the squares of the increments in the case \(\alpha \in (1,2)\) .
The preceding convergence can be equivalently written as
\(\mathopen {}\mathclose {\left(\frac{1}{r_{n}} \mathopen {}\mathclose {\left(S^{n}_{\lfloor n t \rfloor } - mn t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\mathopen {}\mathclose {\left(X^{\alpha ,\lambda ,\mathrm {br}}_{t} -\lambda t ; 0 \leqslant t \leqslant 1\right)},\) 
for the Skorokhod topology. Let us denote by \(X^{\alpha ,\lambda }\)  the process on the right, which is informally the stable process \(X^{\alpha }\)  conditioned on \(X^{\alpha }_{1} = \lambda \) . Note that its jumps are the same as those of \(X^{\alpha ,\lambda ,\mathrm {br}}\) .
This convergence implies that for every \(N\)  fixed, the \(N\)  largest values amongst \(\lbrace r_{n}^{-1} (X^{n}_{k} - m), 1 \leqslant k \leqslant n\rbrace \)  jointly converge towards the values of the \(N\)  largest jumps of \(X^{\alpha ,\lambda }\) . Consequently the sum of the squares of the former converge to that of the latter.
Recall that we assume \(\alpha \in (1,2)\)  so \(\sum _{k=1}^{n} X^{n}_{k} = x_{n} \sim m n = o(r_{n}^{2})\)  and thus for every \(N \geqslant 1\)  fixed, the sum of the largest \(N\)  largest values amongst \(\lbrace r_{n}^{-2} (X^{n}_{k})^{2}, 1 \leqslant k \leqslant n\rbrace \)  converges towards the sum of the \(N\)  largest jumps squared of \(X^{\alpha ,\lambda }\) .
We next argue that one can fix \(N\)  large such that both rests are small with high probability, which implies our claim.
Recall that \(S^{n}\)  has the same law as the random walk \(\widehat{S}\)  conditioned on \(\widehat{S}_{n} = x_{n}\)  so we may replace the \(X^{n}_{k}\) 's by the increments \(\widehat{X}_{k}\) 's of the latter, under the conditional law.
We first notice that this holds true for the unconditioned processes. Indeed it is well-known that setting \((S^{\alpha }_{t})_{t \in [0,1]} = (\sum _{s\leqslant t} (\Delta X^{\alpha }_{t})^2)_{t \in [0,1]}\)  defines a stable subordinator with index \(\alpha /2\) , which is a pure jump process. Moreover, from the tail behaviour of \(\widehat{X}^{2}\) , one can check (using e.g. [2]} and [3]}) that the rescaled process \((r_{n}^{-2} \sum _{k=1}^{{nt}} (\widehat{X}_{k})^{2})_{t \in [0,1]}\)  converges in distribution towards \((S^{\alpha }_{t})_{t \in [0,1]}\) . Note that no centering is needed here since \(\alpha /2 < 1\) .
Then almost surely, \(S^{\alpha }_{1/2} = \sum _{s\leqslant 1/2} \Delta S^{\alpha }_{s}\)  and one can fix \(N\)  such that the sum on the right, when the \(N\)  largest terms are excluded, is arbitrarily small with high probability. Using the preceding convergence, one gets that the same holds for the sequence \((\widehat{X}_{k})_{k \leqslant {n/2}}\) .
Next, these properties are transferred to the bridge conditioning by the absolute continuity with the unconditioned process: There exists \(C>0\)  such that, both for the discrete and the continuum process, the probability of any event which only depends on the first half of the bridge is bounded by \(C\)  times the same probability for the unconditioned process. For \(X^{\alpha }\) , this follows from the fact that the density \(d^{\alpha }_{t}\)  is a bounded function for any \(t>0\) , whereas for \(\widehat{S}\)  one also appeals to the local limit theorem (REF ). Using the invariance of the bridges under space-time reversal, we deduce that for every \(\varepsilon , \delta > 0\) , one can fix \(N\)  such that for every \(n\)  large enough, for each half, the sum of the square of the increments excepted the \(N\)  largest ones are smaller than \(\delta \)  with probability at least \(1-\varepsilon \)  and the proof is complete.
The convergence of the sum of the squares of the increments in the case \(\alpha =2\)  is deferred to Proposition REF .

The condensation phase
Let us finally focus on Theorem REF : Here we assume that \(x_{n}=mn+ \lambda _{n}\)  with \(\lambda _{n}/r_{n} \rightarrow \infty \)  and that \(p(k)\rho ^{-k}\)  is regularly varying at infinity with some index \(-\beta < -2\) .
If \(G^{(2)}(\rho )<\infty \)  (which implies \(\beta \geqslant 3\) ), we assume furthermore that there exists \(c>(\beta -3)/\sigma ^{2}\)  such that \(\lambda _{n} \geqslant \sqrt{c \ln (n)}\) .
For every \(1 \leqslant i \leqslant n\) , let us set \(Y_{i}^{n} = X_{i}^{n} - m\)  and then \(T^{n}_{i} = Y_{1}^{n} + \dots + Y_{i}^{n} = S^{n}_{i} - im\) .
The first claim is equivalent to the convergence
\(\mathopen {}\mathclose {\left(\frac{1}{\lambda _{n}} \cdot T^{n}_{{nt}} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }(\mathbb {1}_{\lbrace U \leqslant t\rbrace }; 0 \leqslant t \leqslant 1),\) 
where \(U\)  is uniformly distributed on \([0,1]\) .
This is a straightforward consequence of a much more precise result.
Precisely, let
\(V_n \inf \mathopen {}\mathclose {\left\lbrace 1\leqslant j \leqslant n : Y^{n}_j=\max \lbrace Y^{n}_i : 1\leqslant i \leqslant n\rbrace  \right\rbrace }\) 
be the first index of the maximal increment of \((T^{n}_{1}, \ldots ,T^{n}_{n})\) .
Let \((Y_{i})_{i \geqslant 1}\)  be a sequence of i.i.d. random variables with distribution given by
\(\mathbb {P}\mathopen {}\mathclose {\left(Y_{1} = k-m\right)} = \frac{1}{G(\rho )} \rho ^{k} p(k)\qquad \text{for every}\quad k \geqslant 0.\) 
The following result is a direct application of [5]}, combined with [6]} (in the case \(G^{(2)}(\rho )<\infty )\)  and [6]} (in the case \(G^{(2)}(\rho )=\infty \) ).
Proposition 4.6 
We have
\(d_{\mathrm {TV}}\mathopen {}\mathclose {\left( \big ( Y^{n}_{1}, \ldots , Y^{n}_{V_{n}-1},Y^{n}_{V_{n}+1}, \ldots , Y^{n}_{n} \big ), \big (Y_{1}, \dots , Y_{n-1}\big ) \right)} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0,\) 
where \(d_{\mathrm {TV}}\)  denotes the total variation distance on \(\mathbb {R}^{n-1}\)  equipped with the product topology.

Theorem REF  is now a simple consequence of this result.
[Proof of Theorem REF ]
Let the \(Y_{i}\) 's be as above; note that \(\max (Y_{1}, \dots ,Y_{n-1})/r_{n}\)  and \((Y_{1} + \dots + Y_{n-1})/r_{n}\)  both converge in distribution. Since we impose \(Y_{1}^{n}+ \dots + Y^{n}_{n}=\lambda _{n}\) , with \(\lambda _{n}/r_{n}\rightarrow \infty \) , then Proposition REF  implies that
\(\frac{Y^{n}_{V_{n}}}{\lambda _{n}} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1\qquad \text{and}\qquad \frac{\max \lbrace Y^{n}_i : i \ne V_{n}\rbrace }{\lambda _{n}} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }0.\) 
Also, by exchangeability of the \(Y^{n}_{k}\) 's, we have \(V_n \rightarrow U\)  in distribution.
This yields (REF ), which is equivalent to the convergence of the processes in Theorem REF .
Next, as in the proof of Theorem REF , the rescaled sum \(r_{n}^{-2} \sum _{k=1}^{n-1} (X_{k})^{2}\)  converges in distribution as \(n \rightarrow \infty \) . The limit is \(\sum _{t \in [0,1]} (\Delta X^{\alpha }_{t})^{2}\)  in the case \(\alpha \in (1,2)\) , and it is constant when \(\alpha =2\) , precisely simply 2 in the case of infinite variance when \(G^{(2)}(\rho ) = \infty \) , and to \(2\mathbb {E}[X_{1}^{2}]/\mathrm {Var}(X_{1})\)  otherwise.
Consequently,
\(\frac{1}{(\lambda _{n})^{2}} \sum _{k \ne V_{n}} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }0,\) 
and the second convergence of Theorem REF  follows since \(X^{n}_{V_{n}}/ \lambda _{n} \rightarrow 1\)  in probability.

Biconditioned random trees
In this section, we focus on random trees with \(n\)  vertices and \(K_{n}\)  leaves sampled proportionally to a sequence of weights. In the case of the uniform distribution, Labarbe & Marckert [13]} proved the convergence of the so-called contour function. We shall here only consider the Łukasiewicz path, which will be sufficient to our application to random maps in the next section. It also provides some combinatorial information, such as the largest degree for example.
In Sec. REF  we first recall the definition of the Łukasiewicz path of a plane tree, then in Sec. REF  we precisely define the distribution we consider. In Sec. REF  we relate the random Łukasiewicz paths to the nondecreasing paths studied in the preceding section. Finally in Sec. REF  we state and prove our main results on scaling limits of these Łukasiewicz paths.
Plane trees and coding path
Recall that a rooted plane tree is a connected graph with no cycle, with a distinguished root-corner, and such that the neighbours of every vertex are ordered. We shall interpret them as genealogical trees; the vertex at the root is the ancestor, the neighbour of a vertex closer to the root is its parent, whereas the other ones are its offspring (ordered from left to right); more generally, the vertices lying between the root and a given vertex are the ancestors of the latter. Finally an individual with no child is called a leaf, the other ones are called internal vertices.
<FIGURE>Let us recall the coding of a plane tree by a discrete path, see Figure REF  for an example.
Fix a tree \(T\)  with \(n\)  vertices; thanks to the planar ordering, we may list these vertices as \(u_0 < u_1 < \dots < u_{n-1}\)  in depth-first search order, which corresponds to the lexicographical order when trees are viewed as words. Then for each such vertex \(u\) , let us denote by \(k_{u} \geqslant 0\)  its offspring number. We define then the Łukasiewicz path \(W(T) = (W_{i}(T) ; 0 \leqslant i \leqslant n)\)  of the tree recursively by \(W_{0}(T) = 0\)  and
\(W_{i+1}(T) = W_{i}(T) + k_{u_i}-1\qquad 0 \leqslant i \leqslant n-1.\) 
One easily checks that \(W_{n}(T) = -1\)  whereas for every \(0 \leqslant i \leqslant n-1\) , we have \(W_{i}(T) \geqslant 0\)  as well as \(W_{i+1}(T)-W_{i}(T) \geqslant -1\) .
There is a 1-to-1 correspondence between the set of such paths and plane trees with \(n\)  vertices, see e.g. [14]} or [15]}. Furthermore the leaves of the tree correspond to the negative increments of its Łukasiewicz path. For every \(k \in \lbrace 0, \dots , n-1\rbrace \) , we shall denote by
\(\Lambda _{k}(T) = \sum _{i = 0}^{k} \mathbb {1}_{\lbrace W_{i+1}(T) - W_{i}(T) = -1\rbrace }\) 
the number of leaves amongst the first \(k+1\)  vertices of \(T\) .

Random trees
We consider random trees sampled according to a sequence of weights.
For a tree \(T\)  and a vertex \(u \in T\) , recall that \(k_{u} \geqslant 0\)  denotes its number of children. Fix a sequence of nonnegative real numbers \({\theta } = (\theta (i))_{i \geqslant 0}\)  which, in order to avoid trivialities, satisfies \(\theta (0) > 0\)  and \(\theta (i) > 0\)  for at least one \(i \geqslant 2\) , and define a measure \(w^{{\theta }}\)  on the set of all finite trees by setting
\(w^{{\theta }}(T) = \prod _{u \in T} \theta (k_{u}),\) 
for every finite tree \(T\) .
For \(n \geqslant K_{n} \geqslant 1\)  let \(\mathbf {T}_{n,K_{n}}\)  denote the finite set of trees with \(n\)  vertices amongst which \(K_{n}\)  are leaves. We shall always implicitly assume that \(n\)  and \(K_{n}\)  are compatible with the support of \({\theta }\) , in the sense that \(\mathbf {T}_{n,K_{n}}\)  has nonzero \(w^{{\theta }}\) -weight. We then define a probability \(\mathbb {P}^{{\theta }}_{n,K_{n}} = w^{{\theta }}(\cdot )/w^{{\theta }}(\mathbf {T}_{n,K_{n}})\)  on this set.
Trees with solely \(n\)  vertices sampled proportionally to their \(w^{{\theta }}\) -weight are known in the literature as simply generated. When the sequence \({\theta }\)  is a probability measure, they correspond to Bienaymé–Galton–Watson trees with offspring distribution \({\theta }\)  conditioned on having \(n\)  vertices; similarly, \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  is the law of such a random tree conditioned to have \(n\)  vertices and \(K_{n}\)  leaves. A classical fact for trees conditioned only by their number of vertices is that an exponential tilting of \({\theta }\)  as in Lemma REF  leaves the distribution of the random trees unaffected, see e.g. the survey of Janson [16]}. Although we shall not need it, let us mention that in the context of trees conditioned both by their number of vertices and leaves, one can gain a degree of freedom by only making the exponential change to the weights \((\theta (i))_{i \geqslant 1}\)  and setting for 0 any arbitrary value.
We next consider the random Łukasiewicz path of such a tree.
For every \(n \geqslant 1\) , let \(\mathbf {B}_{n,K_{n}}\)  denote the set of paths \(w = (w_{k} ; 0 \leqslant k \leqslant n)\)  such that
\(w_{0} = 0, \qquad w_{n} = -1, \qquad w_{k} - w_{k-1} \in \mathbb {Z}_{\geqslant -1} \text{for every}1 \leqslant k \leqslant n,\) 
and
\(\#\lbrace k \in \lbrace 1, \dots , n\rbrace  : w_{k} - w_{k-1} = -1\rbrace  = K_{n}.\) 
Further, let \(\mathbf {W}_{n,K_{n}} \subset \mathbf {B}_{n,K_{n}}\)  denote the set of such paths which in addition satisfy
\(w_{k} \geqslant 0 \qquad \text{for every}\qquad 1 \leqslant k \leqslant n-1.\) 
Let us define probability measures on the sets \(\mathbf {B}_{n,K_{n}}\)  and \(\mathbf {W}_{n,K_{n}}\) , denoted respectively by \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  and \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\) , as in Sec. REF , by normalising the measures which assign a weight
\(\prod _{k=1}^{n} \theta (w_{k}-w_{k-1}+1),\) 
to every path \(w\)  in \(\mathbf {B}_{n,K_{n}}\)  and \(\mathbf {W}_{n,K_{n}}\)  respectively.
Then the coding of a plane tree \(T\)  by its Łukasiewicz path \(W(T)\)  bijectively maps \(\mathbf {T}_{n,K_{n}}\)  onto \(\mathbf {W}_{n,K_{n}}\)  and the law \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\)  on \(\mathbf {W}_{n,K_{n}}\)  is the image measure of \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  on \(\mathbf {T}_{n,K_{n}}\) .

The Vervaat transform
Our aim is to study the asymptotic behaviour of the preceding random Łukasiewicz paths. In order to relax the positivity constraint, and therefore to study paths sampled from \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  instead of \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\) , we make use of the notion of Vervaat transform, also known as cyclic shift, of a bivariate sequence (with respect to the first variable).
First, if \({x}=(x_{k})_{1 \leqslant k \leqslant m}=(a_{k},b_{k})_{1 \leqslant k \leqslant m}\)  is a sequence of integer-valued couples and \(i \in \mathbb {Z}/m\mathbb {Z}\) , we define the cyclic shift \({x}^{(i)}\)  by \(x^{(i)}_k=(a_{k+i \bmod m},b_{k+i \bmod m})\)  for \(1 \leqslant k \leqslant m\)  (where representatives modulo \(m\)  are chosen in \( \lbrace 1,2, \ldots ,m\rbrace \) ). We then define the (discrete) Vervaat transform \( {V}^{\mathsf {d}}({x})\)  as follows (we introduce the superscript \(\mathsf {d}\)  to underline the fact that this transformation acts on discrete sequences). Let \(i_{*}({x})\)  be defined by
\(i_{*}({x}) = \min \mathopen {}\mathclose {\left\lbrace  j \in \lbrace 1,2, \ldots ,m\rbrace  ; a_{1}+a_{2}+\cdots +a_{j}= \min _{1 \leqslant i \leqslant m} (a_{1}+a_{2}+\cdots +a_{i}) \right\rbrace }.\) 
Then \({V}^{\mathsf {d}}({x}) {x}^{(i_{*}({x}))}\) .
Lemma 5.1 
Fix \(n\)  and \(K_{n}\)  and given \(W^{n} \in \mathbf {B}_{n,K_{n}}\) , for all \(k \in \lbrace 0, \dots , n\rbrace \) , let
\(\Lambda ^{n}_{k}=\sum _{i=1}^{k}\mathbb {1}_{\lbrace W^{n}_{i}-W^{n}_{i-1}=-1\rbrace }\) 
denote the number of negative increments up to time \(k\) .
Then the following equality holds in distribution:
\({V}^{\mathsf {d}}\big ((W^{n}_{ k}, \Lambda ^{n}_{k})_{0 \leqslant k \leqslant n}\big ) \text{under}\mathbb {P}^{{\theta }}_{n,K_{n}}\mathop {=}^{(d)}(W^{n}_{ k}, \Lambda ^{n}_{k})_{0 \leqslant k \leqslant n} \text{under}\mathbb {P}^{{\theta },+}_{n,K_{n}}.\) 


This follows from a simple extension of the so-called cyclic lemma (see e.g. [14]}). Indeed, if \({x}=(a_{i},b_{i})_{1 \leqslant i \leqslant m}\)  are integers such that \(a_{1}+a_{2}+\cdots +a_{m}=-1\) , then there is a unique \(j \in \mathbb {Z}/m\mathbb {Z}\)  such that the cyclic shift \({x}^{(j)}=(a^{(j)}_{i},b^{(j)}_{i})_{1 \leqslant i \leqslant m}\)  fulfills \(a^{(j)}_{1}+ \cdots +a^{(j)}_{i} \geqslant 0\)  for every \(1 \leqslant i<m\) . It is then standard to obtain the desired results by an exchangeability argument; we leave details to the reader.
In the continuum setting, following Miermont [18]}, we define the Vervaat transform \(V\, f\)  of a function \(f \in [0,1])\)  with \(f(0)=f(1-)=f(1)\)  as follows:
let \(t_{\min }\)  be the location of the left-most minimum of \(f\) , that is, the smallest \(t\)  such
that \(f(t-) \wedge f (t) = \inf \, f\) , then, for \(t \in [0,1)\)  set
\(V\, f (t) =f(t + t_{\min } \bmod 1]) - \inf _{[0,1]} f,\) 
and \(V\, f (1) = \lim _{t \uparrow 1} V\, f (t)\) .
Note that if \(f \in [0,1])\)  with \(f(0)=f(1-)=f(1)\)  reaches its infimum at a unique time with no jump at that time, then \( {V}\)  is continuous at \(f\) . This is the case of the spectrally positive stable processees with drift from Sec. REF .
Finally, the following simple observation is really the key idea that will allow us to transfer results from Sec.  to the present `biconditioned' setting. Recall from Sec. REF  the law \(\mathbb {P}^{{p},n}_{x_{n}}\)  on nondecreasing paths which end at \(x_{n}\)  at time \(n\) , sampled from a weight sequence \({p}\) .
Lemma 5.2 
Fix a weight sequence \({\theta }\)  and define two sequences \({p}\)  and \({q}\)  by \(p(k) = \theta (k+1)\)  for every \(k\geqslant 0\) , \(q(k) = 1\)  if \(k \in \lbrace 0,1\rbrace \) , and \(q(k) = 0\)  for \(k \geqslant 2\) .
Fix \(1 \leqslant K_{n} \leqslant n\) . Independently, sample \((S^{n-K_{n}}_{k} ; 0 \leqslant k \leqslant n-K_{n})\)  with distribution \(\mathbb {P}^{{p},n-K_{n}}_{K_{n}-1}\)  and \((L^{n}_{k} ; 0 \leqslant k \leqslant n)\)  with distribution \(\mathbb {P}^{{q},n}_{K_{n}}\) . For every \(k \in \lbrace 0, \dots , n\rbrace \) , define
\(W^{n}_{k} = S^{n-K_{n}}_{k - L^{n}_{k}} - L^{n}_{k}.\) 
Then the path \(W^{n}\)  has law \(\mathbb {P}^{{\theta }}_{n,K_{n}}\) .

The proof is straightforward and left to the reader. Intuitively speaking, this just amounts to saying that in a simply generated bridge of \(\mathbf {B}_{n,K_{n}}\) , the position of the \(K_{n}\)  negative steps are uniformly distributed amongst the \(n\)  different jumps and the path obtained by removing them is then a simply generated nondecreasing path which ends at \(K_{n}-1\)  at time \(n-K_{n}\) .

Scaling limits of random Łukasiewicz paths
We are now ready to state and prove several invariance principles for the Łukasiewicz paths of \({\theta }\) -simply generated random trees with \(n\)  vertices and \(K_{n}\)  leaves. Recall from the above discussion that these paths \(W^{n}\)  have the law \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\) . In other words, they are paths starting from 0, whose increments all belong to \(\lbrace -1, 0, 1, 2, \dots \rbrace \) , which reach at time \(n\)  the value \(-1\)  for the first time by making their \(K_{n}\) 'th negative increments, and they are sampled at random proportionally to their weight (REF ). For every \(k \in \lbrace 1, \dots , n\rbrace \) , we shall denote by \(X^{n}_{k} = W^{n}_{k} - W^{n}_{k-1}\)  the \(k\) 'th increment of \(W^{n}\) .
We shall canonically view our discrete paths as càdlàg functions on \([0,1]\) , equipped with the Skorokhod \(J_{1}\)  topology; recall that when the limit is a continuous path, convergence in this topology is equivalent to the uniform convergence. In the following statement, we denote by \(B^{\mathrm {exc}}\)  a standard Brownian excursion, constructed e.g. from the Brownian bridge presented in Sec. REF  via the Vervaat transform as above.
We need to introduce some notation. Set
\(F(z)=\sum _{k=0}^{\infty } \theta (k) z^{k}\qquad \text{and}\qquad A(z)=1- \frac{F(z)-F(0)}{zF^{\prime }(z)}.\) 
Denote by \(\rho \)  the radius of convergence of \(F\)  and set \(k_{0}= \min \lbrace k \geqslant 1: q_{k} \ne 0\rbrace \) ; then it is a simple matter to check that \(A\)  is increasing on \((0, \rho )\)  and we extend it by continuity with \(A(0) = 1-1/k_{0}\)  and \(A(\rho ) < 1\)  if and only if \(F^{\prime }(\rho ) < \infty \) .
The next result is analogous to (and indeed based on) Theorem REF .
Theorem 5.3 
Let \((W^{n}_{k} ; 0 \leqslant k \leqslant n)\)  have the law \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\)  for every \(n \geqslant 1\) .
Then the convergences
\(\mathopen {}\mathclose {\left(\frac{1}{\sqrt{v_{n}}} \cdot W^{n}_{{nt}} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }B^{\mathrm {exc}}\qquad \text{and}\qquad \frac{1}{v_{n}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1,\) 
hold in each of the following cases:
 There exists \(\tau \in (1-1/k_{0},A(\rho ))\)  such that \(K_{n}/n \rightarrow \tau \)  and \(v_{n} = b F^{(2)}(b) n/F^{\prime }(b)\)  with \(b=A^{-1}(\tau )\) .

 \(K_{n}/n \rightarrow 0\)  with \(K_{n} \rightarrow \infty \) , \(\theta (1), \theta (2)>0\) , and \(v_{n} = 2 K_{n}\) .

 \(K_{n}/n \rightarrow 1\)  and \(n-K_{n} \rightarrow \infty \) , \(F\)  is \(\Delta \) -analytic and there exist \(\alpha , c>0\)  such that \(F(\rho -z) \sim c/z^{\alpha }\)  as \(z \rightarrow 0\)  with \(\mathrm {Re}(z)>0\) , and finally \(v_{n} = (1+\alpha )n^{2}/(\alpha (n-K_{n}))\) .
Our last main result on these random paths is analogous to (and again, based on) Theorem REF  and studies specifically the case \(K_{n}/n \rightarrow A(\rho )\)  when the latter is finite. Here, we assume that the probability distribution with generating function \(F(\rho \,\cdot )/F(\rho )\)  belongs to the domain of attraction of a stable law with index \(\alpha \in (1, 2]\) , i.e. that there exist \(\alpha \in (1,2]\) , \(\rho >0\) , \(m>0\) , and a slowly varying function \(L\)  such that for every \(0 \leqslant s < \rho \) ,
\(F(s)=F(\rho ) \mathopen {}\mathclose {\left(1-m+\frac{m s}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{\alpha } L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\right)}.\) 
In the case \(F^{(2)}(\rho ) < \infty \) , i.e. when \(\alpha =2\)  and \(L\)  admits a finite limit at infinity, say \(\ell \) , the law with generating function \(F(\rho \,\cdot )/F(\rho )\)  has a finite variance, equal to
\(\sigma ^{2} = 2\ell + m - m^{2}= \frac{\rho ^{2} F^{(2)}(\rho )}{F(\rho )} + \frac{\rho F^{\prime }(\rho )}{F(\rho )} - \mathopen {}\mathclose {\left(\frac{\rho F^{\prime }(\rho )}{F(\rho )}\right)}^{2}.\) 
In this case we set \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  for every \(n\geqslant 1\) , otherwise, we let \((r_{n})_{n \geqslant 1}\)  be a sequence such that \(r_{n}^{\alpha } \sim n L(r_{n})\)  as \(n \rightarrow \infty \) .
We have already mentioned and used the fact that if \((S_{n})_{n\geqslant 1}\)  is a random walk with i.i.d. increments with generating function \(F(\rho \,\cdot )/F(\rho )\) , then the convergence
\(\mathopen {}\mathclose {\left(\frac{1}{r_{n}} \cdot \mathopen {}\mathclose {\left(S_{{nt}} - mnt\right)} ; t \geqslant 0\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }X^{\alpha },\) 
holds in distribution in \([0,\infty ), \mathbb {R})\) , where \(X^{\alpha }\)  is the \(\alpha \) -stable Lévy process whose law is characterised by \(\mathbb {E}[\exp (-q X^{\alpha }_{t})] = \exp (t q^{\alpha })\)  for every \(q,t > 0\) .
Recall finally from Sec. REF  that for \(\lambda \in \mathbb {R}\) , we denote by \(X^{\alpha ,\lambda ,\mathrm {br}}\)  the process \((X^{\alpha }_{t} - \lambda t)_{t}\)  conditioned to end at 0 at time 1. Let us also denote by \(\Delta X^{\alpha ,\lambda ,\mathrm {br}}_{t} = X^{\alpha ,\lambda ,\mathrm {br}}_{t}-X^{\alpha ,\lambda ,\mathrm {br}}_{t-}\)  the value of the `jump' of \(X^{\alpha ,\lambda ,\mathrm {br}}\)  at time \(t \in [0,1]\) , which is null except for a countable set of times. We let further \(X^{\alpha ,\lambda ,\mathrm {exc}}\)  denote the associated excursion, e.g. obtained after applying the Vervaat transform to \(X^{\alpha ,\lambda ,\mathrm {br}}\) .
Theorem 5.4 
Under the preceding assumptions, if \(K_{n}\)  takes the form
\(K_n = A(\rho ) n + \lambda _{n},\) 
where
\(\lambda _{n}/r_{n}\)  converges to some \(\lambda \in [-\infty ,\infty ]\) , then three cases occur.
 If \(\lambda \in \mathbb {R}\) , then with \(\widehat{\lambda } = \lambda m / (1-A(\rho ))\) ,
\(\mathopen {}\mathclose {\left( \frac{1}{r_{n}} \cdot W^{n}_{{nt}} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\sqrt{1 + \frac{m^{2}-m}{\sigma ^{2}} \mathbb {1}_{\lbrace F^{(2)}(\rho ) < \infty \rbrace }} \cdot \mathopen {}\mathclose {\left(X^{\alpha ,\widehat{\lambda },\mathrm {exc}}_{t/m} ; 0 \leqslant t \leqslant 1\right)}.\) 

In addition,
\(\frac{1}{r_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\begin{dcases}\frac{2 F(\rho )}{\rho F^{\prime }(\rho )} \mathopen {}\mathclose {\left(1 + \frac{m^{2}-m}{\sigma ^{2}} \mathbb {1}_{\lbrace F^{(2)}(\rho ) < \infty \rbrace }\right)} & \text{when } \alpha =2, \\\sum _{t \in [0,1]} (\Delta X^{\alpha ,\widehat{\lambda },\mathrm {exc}}_{t/m})^{2} & \text{when } \alpha \in (1,2).\end{dcases}\) 


 If \(\lambda = -\infty \)  and if \( \lambda _{n}/n \rightarrow 0\) , then with \(\varepsilon _{n}\)  such that \(A(\rho (1-\varepsilon _{n})) = K_{n}/n\) , we have
\(\mathopen {}\mathclose {\left(\sqrt{\frac{\varepsilon _{n}}{|\lambda _{n}|}} \cdot W^{n}_{{nt}} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\sqrt{C} \cdot B^{\mathrm {exc}}\qquad \text{and}\qquad \frac{\varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }C,\) 
where
\(C = \frac{\alpha -1}{1-A(\rho )} \mathbb {1}_{\lbrace F^{(2)}(\rho ) = \infty \rbrace } + \frac{2\ell }{2\ell (1-A(\rho )) - m A(\rho )} \mathbb {1}_{\lbrace F^{(2)}(\rho ) < \infty \rbrace }.\) 

 If \(\lambda = \infty \) , we further assume that \(\theta (k) \rho ^{-k}\)  is regularly varying at infinity with some index \(-\beta < -2\) .
If \(F^{(2)}(\rho )<\infty \)  (then \(\beta \geqslant 3\) ), we assume furthermore that there exists \(c>(\beta -3)/\sigma ^{2}\)  such that \(\lambda _{n} \geqslant \sqrt{c \ln (n)}\) .
Then
\( \mathopen {}\mathclose {\left( \frac{1}{\lambda _{n}} W^{n}_{\max \lbrace 0, {nt}\rbrace } ; -1 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }((1-t) \mathbb {1}_{\lbrace t>0\rbrace }; -1 \leqslant t \leqslant 1).\) 
In addition,
\( \frac{\max \lbrace X^{n}_{1}, \ldots , X^{n}_{n}\rbrace ^{2} }{\sum _{k=1}^{n} (X^{n}_{k})^{2} } \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1.\) 

Let us note that in the last regime, we extend both sides by 0 on the interval \([-1, 0)\) ; this is only because the function \(t\mapsto (1-t) \mathbb {1}_{\lbrace t>0\rbrace }\)  on \([0,1]\)  is not càdlàg. Also, by combining both results, we deduce that \(\max \lbrace X^{n}_{1}, \ldots , X^{n}_{n}\rbrace  / \lambda _{n} \rightarrow 1\)  in probability.
Remark 5.5 
In the case \(F^{(2)}(\rho ) < \infty \) , one can check that in the two cases \(\lambda \in \mathbb {R}\) , or \(\lambda = -\infty \)  and \(\lambda _{n}/n \rightarrow 0\) , we have the following unified convergences:
\(\mathopen {}\mathclose {\left( \frac{1}{\sqrt{n}}\, W^{n}_{{nt}} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\sqrt{\frac{\rho F^{(2)}(\rho )}{F^{\prime }(\rho )}} \, B^{\mathrm {exc}}\quad \text{and}\quad \frac{1}{n} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\frac{\rho F^{(2)}(\rho )}{F^{\prime }(\rho )}.\) 
In other words, Theorem REF  also holds for \(\tau =A(\rho )\)  in this finite variance regime.

In order to lighten the notation, in the proof of these theorems, we assume that all discrete-time processes are extended to continuous time by assigning at any time \(t\)  the same value as at \({t}\) , so we write e.g. \(W_{nt}\)  instead of \(W_{{nt}}\) .
[Proof of Theorem REF ]
Recall that the Brownian excursion \(B^{\mathrm {exc}}\)  can be constructed as the Vervaat transform \( {V} B^{\mathrm {br}}\)  of the Brownian bridge. Since \(B^{\mathrm {br}}\)  attains its infimum almost surely at a unique time, by standard continuity properties of the Vervaat transform, it is enough to show that the scaling limits of paths sampled according to \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  is \(B^{\mathrm {br}}\) .
Generalities.
We let henceforth \(W^{n}\)  denote a random path sampled from \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  and we let \(\Lambda ^{n}\)  be as in (REF ).
Finally, introduce \((S^{n-K_{n}}_{k} ; 0 \leqslant k \leqslant n-K_{n})\)  and \((L^{n}_{k}; 0 \leqslant k \leqslant n)\)  as in Lemma REF , so that the path \((W^{n}_{nt} ; 0 \leqslant t \leqslant 1)\)  has the same law as the process defined for every \(t \in [0,1]\)  by

Sn-Knnt - Lnnt - Lnnt
= (Sn-Knnt - Lnnt - Kn nt - Lnntn-Kn) - (Lnnt - Knt) - Kn (t - nt - Lnntn-Kn)
= (Sn-Knnt - Lnnt - Kn nt - Lnntn-Kn) - nn-Kn (Lnnt - Knt)

.
The process \(L^{n}\)  fits in our framework, but it also has already been studied in the literature. Indeed, it is a simple example of an urn process studied in [19]}. By Theorem 20.7 in this reference, we have the convergence of the normalised process
\(\mathopen {}\mathclose {\left(\frac{L^{n}_{nt} - K_{n} t}{\sqrt{K_{n} (n-K_{n})/n}} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }B^{\mathrm {br}}.\) 
Note that \(\sqrt{K_{n} (n-K_{n})/n} \leqslant \sqrt{n-K_{n}} = o(n-K_{n})\)  since we assume that \(n-K_{n} \rightarrow \infty \) . Consequently
\(\mathopen {}\mathclose {\left(\frac{nt - L^{n}_{nt}}{n-K_{n}} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }(t ; 0 \leqslant t \leqslant 1).\) 
It remains to study \(S^{n-K_{n}}\) . To this end, recall the definition of \(F\)  and \(A\)  from (REF ) and let \({p} = (\theta (k+1))_{k\geqslant 0}\) . If \(G\)  and \(\Psi \)  are defined by (REF ), then for every \(s \in (0,\rho )\) , we have
\(F(s) = F(0) + s G(s)\qquad \text{and}\qquad A(s) = \frac{\Psi (s)}{1+\Psi (s)}.\) 
Regime i.
If \({\theta }\)  and \(K_{n}\)  satisfy the assumptions of Theorem REF , then, according to Theorem REF , it holds
\(\mathopen {}\mathclose {\left(\frac{1}{\sqrt{(n-K_{n}) \sigma ^{2}_{b}}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{(n-K_{n}) t} - K_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }B^{\mathrm {br}},\) 
with \(\Psi (b) = bG^{\prime }(b)/G(b) = \lim _{n} K_{n}/(n-K_{n}) = \tau /(1-\tau )\)  and \(\sigma ^{2}_{b} = \frac{b^{2} G^{(2)}(b)}{G(b)} + \Psi (b) - \Psi (b)^{2}\) , where we recall that \(G\)  is related to \(F\)  by (REF ).
Combined with (REF ) and standard properties of the Skorokhod topology (see e.g. [20]}), we may replace \((n-K_{n})t\)  by \(nt-L^{n}_{nt}\) , so this gives the asymptotic behaviour of the first term in (REF ). Jointly with (REF ), and since \(K_{n}/n \rightarrow \tau \) , this implies that, with \(B^{\mathrm {br},1}\)  and \(B^{\mathrm {br},2}\)  two independent Brownian bridges, we have, as processes,
\(\frac{1}{\sqrt{n}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L^{n}_{nt}} - L^{n}_{nt}\right)}&=& \frac{1}{\sqrt{n}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L^{n}_{nt}} - K_{n} \frac{nt - L^{n}_{nt}}{n-K_{n}}\right)} - \sqrt{\frac{n}{(n-K_{n})^{2}}} \mathopen {}\mathclose {\left(L^{n}_{nt} - K_{n}t\right)}\\&\displaystyle \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }& \sqrt{(1-\tau ) \sigma ^{2}_{b}}\cdot B^{\mathrm {br},1} - \sqrt{\frac{\tau }{1-\tau }}\cdot B^{\mathrm {br},2}\\&\displaystyle \mathop {=}^{(d)}& \sqrt{(1-\tau ) \sigma ^{2}_{b} + \frac{\tau }{1-\tau }} \cdot B^{\mathrm {br}}.\) 
Notice that \(\tau = \Psi (b)/(1+\Psi (b))\) , it is then a simple matter to check that
\((1-\tau ) \sigma ^{2}_{b} + \frac{\tau }{1-\tau } = \frac{b^{2} G^{(2)}(b) + 2bG^{\prime }(b)}{G(b) + bG^{\prime }(b)} = \frac{b F^{(2)}(b)}{F^{\prime }(b)}.\) 
The first claim in Theorem REF  finally follows from Lemma REF  and the Vervaat transform. Similarly, recall that the second claim is about the limit of \(n^{-1} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  under \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\) ; since this quantity is invariant under cyclic shift, it has the same law under \(\mathbb {P}^{{\theta }}_{n,K_{n}}\) . Notice that the total contribution of the negative increments to the sum is \(K_{n}\) , then we infer from Lemma REF  and Theorem REF  that
\(n^{-1} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }(1-\tau ) \frac{b^{2} G^{(2)}(b) + bG^{\prime }(b)}{G(b)} + \tau = \frac{b F^{(2)}(b)}{F^{\prime }(b)}.\) 
Regime ii.
Theorem REF  can be similarly obtained: we may now combine Theorem REF  with (REF ) and [20]} to deduce the convergence in distribution
\(\mathopen {}\mathclose {\left(\frac{1}{\sqrt{K_{n}}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - K_{n}t} - K_{n} t\right)}; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }B^{\mathrm {br}},\) 
which, together with (REF ), (REF ), (REF ), and the fact that \(K_{n}/n \rightarrow 0\) , yields
\(\mathopen {}\mathclose {\left(\frac{1}{\sqrt{K_{n}}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L_{nt}} - L^{n}_{nt}\right)} ; 0 \leqslant t \leqslant 1\right)}\mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }B^{\mathrm {br},1} - B^{\mathrm {br},2}\mathop {=}^{(d)}\sqrt{2} \cdot B^{\mathrm {br}}.\) 
Furthermore, since the total contribution of the negative increments to \(\sum _{k=1}^{n} (X^{n}_{k})^{2}\)  equals \(K_{n}\) , then Theorem REF  yields
\(\frac{1}{K_{n}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1+1=2.\) 
Regime iii.
As for Theorem REF , we combine Theorem REF  with (REF ) and [20]} to obtain
\(\mathopen {}\mathclose {\left(\sqrt{\frac{n-K_{n}}{K_{n}^{2}}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - K_{n}t} - K_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\frac{1}{\sqrt{\alpha }} \cdot B^{\mathrm {br}},\) 
which, together with (REF ), (REF ), (REF ), and the fact that \(K_{n}/n \rightarrow 1\) , yields
\(\mathopen {}\mathclose {\left(\sqrt{\frac{n-K_{n}}{K_{n}^{2}}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L_{nt}} - L^{n}_{nt}\right)} ; 0 \leqslant t \leqslant 1\right)}\mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\frac{1}{\sqrt{\alpha }} \cdot B^{\mathrm {br},1} - B^{\mathrm {br},2}\mathop {=}^{(d)}\sqrt{\frac{\alpha +1}{\alpha }} \cdot B^{\mathrm {br}}.\) 
And finally, since the total contribution of the negative increments to \(\sum _{k=1}^{n} (X^{n}_{k})^{2}\)  equals \(K_{n} = o(K_{n}^{2}/(n-K_{n}))\) , then by Theorem REF ,
\(\frac{n-K_{n}}{K_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\frac{\alpha +1}{\alpha }.\) 
In both cases, the conclusion follows from Lemma REF  and the Vervaat transform.
The same argument applies also to prove Theorem REF . We shall use the same notation.
[Proof of Theorem REF ]
Recall from Sec. REF  the construction of a bridge \(X^{\mathrm {br}}\)  of \(X\) , a Lévy process with linear drift, we denote by \(X^{\mathrm {exc}} = {V} X^{\mathrm {br}}\)  the associated excursion. The Lévy bridges which appear here are those of stable processes with a drift, their infimum is almost surely attained at a unique time and with no jump at that time.
This can be easily derived from the absolute continuity relation (REF ). See also [23]} for more general results.
Therefore, by standard continuity properties of the Vervaat transform, the scaled distributional convergence of a random path under \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\)  towards \(X^{\mathrm {exc}}\)  will follow from the scaled distributional convergence of the path under \(\mathbb {P}^{{\theta }}_{n,K_{n}}\)  towards \(X^{\mathrm {br}}\) .
Generalities.
By the assumption on \(F\) , the function \(G\)  defined by (REF ) is such that for every \(s \in (0,1)\) :
\(\frac{G(\rho s)}{G(\rho )} = 1 - \frac{L(1)}{m-L(1)} (1-s) + (1-s)^{\alpha } \widehat{L}\mathopen {}\mathclose {\left(\frac{1}{1-s}\right)},\) 
where
\(\widehat{L}(x) = \frac{L(x) - L(1) x^{-(2-\alpha )}}{\frac{x-1}{x} (m-L(1))}.\) 
The function \(\widehat{L}\)  is slowly varying at infinity so \(G\)  satisfies the assumption of Theorem REF . Note that
\(A(\rho ) = \frac{L(1)}{m}\qquad \text{and}\qquad \frac{L(1)}{m-L(1)} = \frac{\rho G^{\prime }(\rho )}{G(\rho )} = \Psi (\rho ) = \frac{A(\rho )}{1-A(\rho )}.\) 
Also, if \(\alpha = 2\)  and \(L\)  tends to \(\ell < \infty \) , then \(\lim _{\infty } \widehat{L} = (\ell - L(1))/(m-L(1)) \widehat{\ell }\) , otherwise \(\widehat{L}(x) \sim L(x)/(m-L(1))\)  as \(x\rightarrow \infty \) .
Recall that in the former case, we let \(r_{n} = \sqrt{n\sigma ^{2}/2}\) , where \(\sigma ^{2} = 2\ell + m - m^{2}\) , whereas in the second case, \(r_{n}\)  is such that \(r_{n}^{\alpha } \sim n L(r_{n})\) . Define similarly \(\widehat{r}_{n} = \sqrt{n\widehat{\sigma }^{2}/2}\)  with \(\widehat{\sigma }^{2} = 2\widehat{\ell } + \Psi (\rho ) - \Psi (\rho )^{2}\)  in the first case, and let \(\widehat{r}_{n}\)  be such that \(\widehat{r}_{n}^{\alpha } \sim n \widehat{L}(\widehat{r}_{n})\)  in the second case.
Since \(K_{n}/n \rightarrow A(\rho )\)  and \(\widehat{\ell } + \Psi (\rho ) = \ell /(m-L(1))\) ,
if \(\alpha = 2\)  and \(L\)  tends to \(\ell < \infty \) , then as \(n\rightarrow \infty \) ,
\(\frac{\widehat{r}_{n-K_{n}}}{r_{n}}\mathop {\longrightarrow }^{}_{n \rightarrow \infty }\sqrt{\frac{(1-A(\rho )) \widehat{\sigma }^{2}}{\sigma ^{2}}}= \sqrt{\frac{1}{\sigma ^{2}} \mathopen {}\mathclose {\left(\frac{2\widehat{\ell } + 2\Psi (\rho )}{1+\Psi (\rho )} - \Psi (\rho )\right)}}= \sqrt{\frac{1}{\sigma ^{2}} \mathopen {}\mathclose {\left(\frac{2\ell }{m} - \Psi (\rho )\right)}},\) 
and otherwise
\(\widehat{r}_{n-K_{n}}\mathop {\sim }^{}_{n \rightarrow \infty }\frac{1}{m^{1/\alpha }} r_{n}.\) 
Finally, let
\(\widehat{\lambda }_{n-K_{n}} = {K_{n} - \Psi (\rho ) (n-K_{n})}= \frac{K_{n} - A(\rho ) n}{1-A(\rho ) }= \frac{\lambda _{n}}{1-A(\rho ) }.\) 
Regime i.
Assume that \(K_{n} = A(\rho )n + \lambda _{n}\)  with \(\lambda _{n}/r_{n} \rightarrow \lambda \in \mathbb {R}\)  and define \(\widehat{\lambda } = \lim _{n} \widehat{\lambda }_{n-K_{n}}/\widehat{r}_{n-K_{n}}\) ,
then Theorem REF  implies
\(\mathopen {}\mathclose {\left(\frac{1}{\widehat{r}_{n-K_{n}}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt-K_{n}t} - K_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }X^{\alpha ,\widehat{\lambda },\mathrm {br}}.\) 
Suppose first that \(\alpha =2\)  and that the function \(L\)  has a finite limit; in this case, the scaling satisfies \(r_{n} \sim \sqrt{n \sigma ^{2}/2}\)  and the preceding convergence, combined with (REF ), (REF ), (REF ), and the fact that \(K_{n}/n \rightarrow A(\rho )\) , yields
\(\mathopen {}\mathclose {\left(\frac{1}{r_{n}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L_{nt}} - L^{n}_{nt}\right)} ; 0 \leqslant t \leqslant 1\right)}\mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\sqrt{\frac{1}{\sigma ^{2}} \mathopen {}\mathclose {\left(\frac{2\ell }{m} - \Psi (\rho )\right)}} \cdot X^{2,\widehat{\lambda },\mathrm {br}} - \sqrt{\frac{2 \Psi (\rho )}{\sigma ^{2}}} \cdot B^{\mathrm {br}},\) 
where \(X^{2,\widehat{\lambda },\mathrm {br}}\)  and \(B^{\mathrm {br}}\)  are independent.
Recall that \(X^{2,\widehat{\lambda },\mathrm {br}}\)  has the same law as \(\sqrt{2} \cdot B^{\mathrm {br}}\) , so the right-hand side above as the same law as
\(\sqrt{\frac{2\ell }{\sigma ^{2}}} \cdot X^{2,\widehat{\lambda },\mathrm {br}}= \sqrt{\frac{\sigma ^{2}-m+m^{2}}{\sigma ^{2}}} \cdot X^{2,\widehat{\lambda },\mathrm {br}}.\) 
Let us next turn to the limit of \(r_{n}^{-2} \sum _{k=1}^{n} (X^{n}_{k})^{2}\) , still in the regime \(\alpha =2\)  and when \(L\)  tends to \(\ell \) . Note that

2
= 2  + () - ()2
= 2 (m (()+1) - ()) + () - ()2
= (2m - ()) (()+1)
.
Since the total contribution of the negative increments to the sum is \(K_{n} \sim A(\rho ) n\) , then we infer from Lemma REF  and Theorem REF  that
\(\frac{1}{r_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2}\mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\frac{1}{\sigma ^{2}} \mathopen {}\mathclose {\left(\frac{2\ell }{m} - \Psi (\rho )\right)} \mathopen {}\mathclose {\left(2+\frac{2\Psi (\rho )^{2}}{\widehat{\sigma }^{2}}\right)} + \frac{2 A(\rho )}{\sigma ^{2}}= \frac{4\ell }{m \sigma ^{2}}= \frac{2 \rho F^{(2)}(\rho )}{F^{\prime }(\rho ) \sigma ^{2}}.\) 
The cases \(\alpha =2\)  and \(\lim _{\infty } L = \infty \)  as well as \(\alpha \in (1,2)\)  are similar, except that now \(\sqrt{n} = o(r_{n})\) , thus the contribution of \(L^{n}_{nt} - K_{n} t\)  in (REF ) is negligible and we simply obtain from Theorem  REF
\(\mathopen {}\mathclose {\left(\frac{1}{r_{n}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L_{nt}} - L^{n}_{nt}\right)} ; 0 \leqslant t \leqslant 1\right)}\mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }m^{-1/\alpha } X^{\alpha ,\widehat{\lambda },\mathrm {br}}.\) 
Note that \(m^{-1/\alpha } \widehat{\lambda } = \lambda / (1-A(\rho ))\) ; we infer from the scaling property of \(X^{\alpha }\)  that the right-hand side has the same law as
the process \((X^{\alpha }_{t/m} - \frac{\lambda m}{1-A(\rho )} \frac{t}{m} ; 0 \leqslant t \leqslant 1)\)  conditioned to be at 0 at time 1.
The convergence of \(r_{n}^{-2} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  again follows from Theorem REF : in the case \(\alpha =2\) , we directly read
\(\frac{1}{r_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2}\mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\frac{2}{m}= \frac{2 F(\rho )}{\rho F^{\prime }(\rho )},\) 
whereas in the case \(\alpha \in (1,2)\) ,
\(\frac{1}{r_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2}\mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }m^{-2/\alpha } \sum _{t \in [0,1]} (\Delta X^{\alpha ,\widehat{\lambda }}_{t})^{2}\mathop {=}^{(d)}\sum _{t \in [0,1]} (\Delta X^{\alpha ,\lambda ,m}_{t/m})^{2}.\) 
In any case we conclude the proof of Theorem REF  by applying Lemma REF  and the Vervaat transform.
Regime ii.
Next, in Theorem REF , recall that we assume \(\lambda _{n}/r_{n} \rightarrow -\infty \)  with \(\lambda _{n}/n \rightarrow 0\) , and we take \(\varepsilon _{n}\)  such that \(A(\rho (1-\varepsilon _{n})) = K_{n}/n\) .
Observe that \(\varepsilon _{n}\)  satisfies \(\Psi (\rho (1-\varepsilon _{n})) = K_{n}/(n-K_{n})\) ; we infer from Theorem REF  that
\(\mathopen {}\mathclose {\left(\sqrt{\frac{\varepsilon _{n}}{|\widehat{\lambda }_{n-K_{n}}| }} \cdot \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt} - K_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\sqrt{\alpha -1} \cdot B^{\mathrm {br}}.\) 
Observe that \(\widehat{\lambda }_{n-K_{n}} \sim \lambda _{n} / (1-A(\rho ))\)  and recall from (REF ) that, when both \(\alpha =2\)  and \(L\)  has a finite limit, then so does \(\widehat{L}\)  and so
\(\frac{|\widehat{\lambda }_{n-K_{n}}|}{(n-K_{n}) \varepsilon _{n}} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }\widehat{\sigma }^{2}.\) 
Combined with (REF ), (REF ), (REF ), we infer in this case that, with \(B^{\mathrm {br},1}\)  and \(B^{\mathrm {br},2}\)  two independent Brownian bridges,
\(\mathopen {}\mathclose {\left(\sqrt{\frac{(1-A(\rho )) \varepsilon _{n}}{|\lambda _{n}|}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L_{nt}} - L^{n}_{nt}\right)} ; 0 \leqslant t \leqslant 1\right)}&\displaystyle \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }& B^{\mathrm {br},1} - \sqrt{\frac{\Psi (\rho )}{(1-A(\rho )) \widehat{\sigma }^{2}}} \cdot B^{\mathrm {br},2}\\&\displaystyle \mathop {=}^{(d)}& \sqrt{1+\frac{\Psi (\rho )}{\frac{2\ell }{m} - \Psi (\rho )}} \cdot B^{\mathrm {br}}\\&=&\sqrt{\frac{2\ell }{2\ell - m \Psi (\rho )}} \cdot B^{\mathrm {br}}.\) 
As for \(r_{n}^{-2} \sum _{k=1}^{n} (X^{n}_{k})^{2}\) , still in the regime \(\alpha =2\)  and when \(L\)  tends to \(\ell \) , since the total contribution of the negative increments to the sum is \(K_{n} \sim A(\rho ) n\) , then we infer from Lemma REF  and Theorem REF  and (REF ) that
\(\frac{(1-A(\rho )) \varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (X^{n}_{k})^{2}\mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1 + \frac{\Psi (\rho )^{2}}{\widehat{\sigma }^{2}} + \frac{\Psi (\rho )}{\widehat{\sigma }^{2}}= \frac{2(\widehat{\ell } + \Psi (\rho ))}{\widehat{\sigma }^{2}}= \frac{2\ell }{2\ell - m \Psi (\rho )}.\) 
Now when either \(\alpha \in (1,2)\)  or \(L\rightarrow \infty \) , (so \(\widehat{L} \rightarrow \infty \) ), recall from (REF ) that
\(\frac{|\lambda _{n}|}{(1-A(\rho )^{2} n \varepsilon _{n}}\mathop {\sim }^{}_{n \rightarrow \infty }\frac{|\widehat{\lambda }_{n-K_{n}}|}{(n-K_{n}) \varepsilon _{n}}\mathop {\sim }^{}_{n \rightarrow \infty }\alpha \varepsilon _{n}^{\alpha -2} \widehat{L}(1/\varepsilon _{n}),\) 
which tends to infinity. Therefore the contribution of \(L^{n}_{nt} - K_{n} t\) , which is of order \(\sqrt{n}\) , is again negligible and we simply obtain
\(\mathopen {}\mathclose {\left(\sqrt{\frac{\varepsilon _{n}}{|\lambda _{n}|}} \mathopen {}\mathclose {\left(S^{n-K_{n}}_{nt - L_{nt}} - L^{n}_{nt}\right)} ; 0 \leqslant t \leqslant 1\right)}\mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }\sqrt{\frac{\alpha -1}{1-A(\rho )}} \cdot B^{\mathrm {br}}.\) 
Further, we read from Theorem REF  that
\(\frac{\varepsilon _{n}}{|\widehat{\lambda }_{n-K_{n}}|} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\alpha -1,\) 
and since now \(K_{n}\)  is small compared to \(\varepsilon _{n} / (|\lambda _{n}|)\) , then
\(\frac{\varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (X^{n}_{k})^{2}\mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\frac{\alpha -1}{1-A(\rho )}.\) 
In any case we conclude the proof of Theorem REF  by applying Lemma REF  and the Vervaat transform.
Regime iii.
For the last regime, recall that \(\widehat{\lambda }_{n-K_{n}} \sim \lambda _{n} / (1-A(\rho ))\)  is large compared to \(\sqrt{n}\) , so the contribution of \(L^{n}\)  in the decomposition of \(W^{n}\)  is negligible, and we infer from Theorem REF  and Lemma REF  that, under \(\mathbb {P}^{{\theta }}_{n,K_{n}}\) ,
\(\mathopen {}\mathclose {\left( \frac{1-A(\rho )}{\lambda _{n}} \cdot \mathopen {}\mathclose {\left(W^{n}_{nt} - K_{n} t\right)} ; 0 \leqslant t \leqslant 1\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }(\mathbb {1}_{\lbrace U \leqslant t\rbrace } - t; 0 \leqslant t \leqslant 1),\) 
Then the first claim in Theorem REF  follows from the Vervaat transform.
As for the second claim, we also deduce from the proof of Theorem REF  that if we let \(U_n \inf \lbrace 1\leqslant j \leqslant n : X^{n}_j=\max \lbrace |X^{n}_i| : 1\leqslant i \leqslant n\rbrace \rbrace \) , then
\(\frac{X^{n}_{U_{n}}}{\widehat{\lambda }_{n-K_{n}} } \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1,\qquad \text{and}\qquad \frac{1}{(\widehat{\lambda }_{n-K_{n}} )^{2}} \sum _{k \ne U_{n}} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }0,\) 
and the second claim of Theorem REF  follows.
Random planar maps
This last section is devoted to the study of random biconditioned Boltzmann planar maps. Let us start in Sec. REF  by precisely defining the model.
We then present in Sec. REF  the bijection between maps and labelled trees and describe the law of the random trees which code Boltzmann maps.
Finally our main results are stated and proved in Sec. REF  by relying on the results from the preceding section.
Figure REF  summarises our results in the different regimes.
<FIGURE>Biconditioned Boltzmann maps
A natural generalisation of the model of random quadrangulations is that of Boltzmann random maps which allows the face degrees to be random, introduced in [24]}; it also naturally appears when considering random maps coupled with statistical physics models, see e.g. [25]}. Specifically, denote by \(\mathbf {M}\)  the set of all finite rooted bipartite planar maps, in which all faces have even degree. Let us fix a sequence of nonnegative real numbers \({q}= (q_i ; i \geqslant 1)\)  which, in order to avoid trivialities, satisfies \(q_i > 0\)  for at least one \(i \geqslant 2\) . We define a measure \(w^{{q}}\)  on \(\mathbf {M}\)  by setting
\(w^{{q}}(M) = \prod _{f \text{ face}} q_{\mathrm {deg}(f)/2},\qquad M \in \mathbf {M},\) 
where \(\mathrm {deg}(f)\)  is the degree of the face \(f\) . When \(w^{{q}}(\mathbf {M}) = \sum _{M \in \mathbf {M}} w^{{q}}(M)<\infty \)  (one says that \({q}\)  is admissible), one can sample a planar map \(M^{{q}}\)  at random proportionally to its \(w^{{q}}\) -weight. The size of \(M^{{q}}\)  is also random but one can condition \(M^{{q}}\)  to have a given number \(n\)  either of vertices, or of edges, or of faces and study its asymptotic behaviour as \(n \rightarrow \infty \) .
Le Gall [26]}, based on [24]}, proved that if, loosely speaking, the weight sequence \({q}\)  is such that the degree of a typical face of \(M^{{q}}\)  has finite exponential moments, then \(M^{{q}}\) , conditioned to have \(n\)  vertices, and rescaled by a factor of order \(n^{-1/4}\)  converges in distribution to the Brownian map.
On the other hand, when the weight sequence \({q}\)  is such that, roughly speaking, a typical face of \(M^{{q}}\)  has degree \(2k\)  with probability of order \(c k^{-1-\alpha }\)  when \(k \rightarrow \infty \) , with \(c>0\)  and \(\alpha \in (1,2)\) , then Le Gall & Miermont [25]} proved a tightness result: From every sequence of integers, one can extract a subsequence along which the map \(M^{{q}}\)  conditioned to have \(n\)  vertices, rescaled by a factor \(n^{-1/(2\alpha )}\) , converges in distribution towards a limit space which has almost surely Hausdorff dimension \(2\alpha \) .
Both results were extended in [29]} to the full domain of attraction of a stable law with index in \((1,2]\) , and by conditioning with respect to either the number vertices, or of edges, or of faces.
Recall from the introduction that we are interested in maps conditioned on having a large fixed number of vertices, edges, and faces at the same time.
More precisely, we consider a sequence \((K_{n})_{n\geqslant 1}\)  of integers and, in order to discard degenerate cases, we shall always assume that both \(K_{n}\)  and \(n-K_{n}\)  tend to infinity. Let us denote by \(\mathbf {M}_{n,K_{n}}\)  the set of all rooted bipartite planar maps with \(n-1\)  edges and \(K_{n}+1\)  vertices; by Euler's formula, all maps in \(\mathbf {M}_{n,K_{n}}\)  have \(n-K_{n}\)  faces (hence the name biconditioned). As before, a weight sequence \({q}= (q_i ; i \geqslant 1)\)  being given, we shall always implicitly assume that \(n\)  and \(K_{n}\)  are compatible with the support of \({q}\) , in the sense that \(w^{{q}}(\mathbf {M}_{n,K_{n}}) > 0\) ; we also assume without further notice that \( \lbrace i \geqslant 1: q_{i}>0\rbrace \)  is not included in a sublattice of \(\mathbb {Z}\) , i.e. the largest \(h>0\)  such that there exists \(a \in \mathbb {R}\)  such that \(\lbrace i \geqslant 1: q_{i}>0\rbrace  \subset a+h\mathbb {Z}\)  is \(h=1\) . The results carry through without such an aperiodicity condition after mild adaptations.
We then define a probability
\(\mathbb {P}^{{q}}_{n,K_{n}}(M)= \frac{w^{{q}}(M)}{w^{{q}}(\mathbf {M}_{n,K_{n}})}, \qquad M \in \mathbf {M}_{n,K_{n}},\) 
so that \(\mathbb {P}^{{q}}_{n,k}\)  is the law of a \({q}\) -Boltzmann planar map, biconditioned to have \(n-1\)  edges and \(K_{n}+1\)  vertices (and \(n-K_{n}\)  faces). Observe that \(\mathbf {M}_{n,K_{n}}\)  is finite, so \(w^{{q}}(\mathbf {M}_{n,K_{n}}) < \infty \)  and we do not need to restrict to admissible weight sequences.
Note that if \(q_{i} = 1\)  for every \(i \geqslant 1\) , then \(\mathbb {P}^{{q}}_{n,K_{n}}\)  is simply the uniform distribution on \(\mathbf {M}_{n,K_{n}}\) ; more generally, if \(A\)  is a subset of integers and \(q_{i}=\mathbb {1}_{\lbrace i \in A\rbrace }\) , then \(\mathbb {P}^{{q}}_{n,K_{n}}\)  is the uniform distribution on the subset of maps of \(\mathbf {M}_{n,K_{n}}\)  having all face degrees in \(2A\) .
In the same way stable processes are the only possible scaling limits of appropriately rescaled random walks, it is natural to expect that the Brownian map and the stable maps are the only possible scaling limits of appropriately rescaled size-conditioned Boltzmann planar maps.
However, as we saw in the preceding sections, the biconditioning in a sense amounts to let the face degree distribution vary with the size, which allows more general scaling limits, in the same way Lévy processes are scaling limits of appropriately rescaled random walks with varying jump distribution. In our case, the Lévy processes that appear are just stable processes with a linear drift.

Bijection with labelled trees
A powerful tool to prove scaling limits of random maps is a coding with labelled trees.
Recall the formal definition of (rooted plane) tree from Sec. . A labelling of such tree \(T\)  is a function \(\ell : V(T) \rightarrow \mathbb {Z}\)  which equips each vertex with an integer, which we can view as a spatial position of the corresponding individual. We say that a labelled tree is a well-labelled tree if it satisfies the following two constraints:

The root of the tree has label 0;

For every internal vertex, say \(u_0\) , with \(k \geqslant 1\)  offspring, say \(u_1\) , ..., \(u_k\)  from left to right, it holds that \(\ell (u_i) \geqslant \ell (u_{i-1})-1\)  for every \(i = 1, \dots , k\) , and \(\ell (u_k) = \ell (u_0)\) .

Observe that since the root has label 0, the label of a vertex is the sum of the label increments between every two successive ancestors so the quantities \(\Xi _{u_{0}} = (0, \ell (u_{1})-\ell (u_{0}), \dots , \ell (u_{k})-\ell (u_{0}))\)  for each internal vertex \(u_{0}\)  entirely describe the labelling \(\ell \) ; finally, the second property states that for every such vertex \(u_{0}\) , the vector \(\Xi _{u_{0}}\)  belongs to the following set of bridges:
\({B}^{\geqslant -1}_{k} = \mathopen {}\mathclose {\left\lbrace (b_{i})_{0 \leqslant i \leqslant k} : b_{0} = 0 = b_{k} \text{ and } b_{i}-b_{i-1} \in \mathbb {Z}_{\geqslant -1} \text{ for } 1 \leqslant i \leqslant k\right\rbrace }.\) 
By combining the works [30]} and [31]}, there is a (constructive) 2-to-1 correspondence between well-labelled trees and bipartite planar maps with a distinguished vertex (in addition to the root-edge); the 2-to-1 comes from the fact that it looses the orientation of the root-edge.
See Figure REF  for an example and [29]} for details.
This correspondence enjoys the following properties:

The edges of the tree correspond to those of the map.

The internal vertices of the tree correspond to the faces of the map and the offspring number of an internal vertex is half the degree of the associated face.

The leaves of the tree correspond to the non-distinguished vertices of the map.

If one shifts all the labels so that the minimum is 1 (one then recovers the original labels by shifting them so the root has label 0), then the label of a leaf equals the distance in the map of the corresponding vertex to the distinguished one.

Observe that by the properties (i) and (iii) above, the labelled tree associated with a bipartite planar map having \(n-1\)  edges and \(k+1\)  vertices with one distinguished vertex has \(n\)  vertices and \(k\)  leaves.
The last property shows that controlling the labels on the tree is equivalent to controlling distances to the distinguished vertex in the map. It then remains to control all the pairwise distances, which is particularly challenging and was done for quadrangulations, by Le Gall [26]} and Miermont [34]}. Thanks to their work, going from the control of labels on the tree to that of all distances in the map, in the case of the Brownian map, is now rather simple.
We will in fact not consider the labels here, but only the tree itself, and simply rely on general results from [35]}.
<FIGURE>Recall the law \(\mathbb {P}^{{q}}_{n,K_{n}}\)  on maps with \(n-1\)  edges and \(K_{n}+1\)  vertices, induced by a sequence of weights \({q}= (q_i ; i \geqslant 1)\) . Property (ii) above allows us to determine the law of the associated random labelled tree.
In the context of the bijection from [30]} only, this was first discussed by Marckert & Miermont [24]}, and it was then adapted using bijection from [31]} in [39]}, see the proof of Proposition 11 in the latter reference.
Recall the set of bridges \({B}_{k}\)  as defined in (REF ); its cardinality is the binomial factor \(\binom{2k-1}{k-1}\) . Then define from \({q}\)  a weight sequence \({\theta } = (\theta (i) ; i \geqslant 0)\)  by
\(\theta (0)=1 \qquad \text{and}\qquad \theta (i) = \binom{2i-1}{i-1} q_{i} \qquad \text{for every }i \geqslant 1.\) 
The preceding correspondence implies that the law of the labelled tree \((T, \ell )\)  associated with a random map \(M\)  sampled from \(\mathbb {P}^{{q}}_{n,K_{n}}\) , in which we distinguish a vertex chosen uniformly at random amongst the \(K_{n}+1\)  possibilities and independently of \(M\) , is as follows: First \(T\)  is a simply generated tree with \(n\)  vertices and \(K_{n}\)  leaves sampled from the weight sequence \({\theta }\)  as defined in Section REF , and then, conditionally given \(T\) , the labels are obtained by sampling independently for every internal vertex \(u\)  a uniform random bridge in \({B}_{k_{u}}\)  as defined in (REF ).

Scaling limits
Our aim is to determine the asymptotic behaviour of a \({q}\) -Boltzmann random planar map \(M_{n,K_n}\)  conditioned to have \(n-1\)  edges and \(K_{n}+1\)  vertices, sampled from \(\mathbb {P}^{{q}}_{n,K_n}\) , and more precisely of the metric measured space obtained by endowing the set of vertices \(V(M_{n,K_n})\)  with the graph distance \(d_{\mathrm {gr}}\) , suitably rescaled, and the uniform probability measure \(p_{\mathrm {unif}}\) , in the Gromov–Hausdorff–Prokhorov topology (see e.g. [40]} for details on this topology).
We need to introduce some notation. Set
\(F(z)=1+\sum _{k=1}^{\infty } \binom{2k-1}{k-1} q_{k} z^{k}\qquad \text{and}\qquad A(z)=1- \frac{F(z)-F(0)}{zF^{\prime }(z)}.\) 
Denote by \(\rho \)  the radius of convergence of \(F\)  and set \(k_{0}= \min \lbrace k \geqslant 1: q_{k} \ne 0\rbrace \) ; then it is a simple matter to check that \(A\)  is increasing on \((0, \rho )\)  and we extend it by continuity with \(A(0) = 1-1/k_{0}\)  and \(A(\rho ) < 1\)  if and only if \(F^{\prime }(\rho ) < \infty \) .
These are the functions defined in (REF ) with the sequence \({\theta }\)  from (REF ).
Example 6.1 
It is instructive to keep in mind the example of uniform bipartite maps, which corresponds to the weight sequence \(q_{i}=1\)  for every \(i \geqslant 1\) . In this case, \(k_{0}=1\) , \(\rho =1/4\) , and
\(F(z)= \frac{1}{2} + \frac{1}{2\sqrt{1-4z}},\qquad \text{and}\qquad A(z)= \frac{6z-1+(1-4z)^{3/2}}{2z}.\) 
One can easily apply the next theorem to deduce Theorem REF  appearing the introduction; for example, the scaling function \(S\)  in the latter is \(S(x) = \frac{F^{\prime }(A^{-1}(x))}{A^{-1}(x) F^{(2)}(A^{-1}(x))}\)  in Theorem REF  below.
The assumptions of Theorem REF  are also satisfied, with \(\alpha = 1/2\) .

Extracting a subsequence if necessary, we may assume that \(K_{n}/n\)  converges as \(n \rightarrow \infty \)  to some limit \(\tau \in [0,1]\) .
Theorem REF  combined with [35]} will readily entail the following result.
Theorem 6.2 
Let \(M_{n,K_n}\)  denote a \({q}\) -Boltzmann random planar map conditioned to have \(n-1\)  edges and \(K_{n}+1\)  vertices.
If \(({M}, {D}, {m})\)  denotes the Brownian map, then the convergence in distribution
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), v_{n}^{-1/4} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }({M}, {D}, {m})\) 
holds for the Gromov–Hausdorff–Prokhorov topology
in each of the following cases:
 There exists \(\tau \in (1-1/k_{0},A(\rho ))\)  such that \(K_{n}/n \rightarrow \tau \)  and \(v_{n} = \frac{4 b F^{(2)}(b) n}{9 F^{\prime }(b)}\)  with \(b=A^{-1}(\tau )\) .

 \(K_{n}/n \rightarrow 0\)  with \(K_{n} \rightarrow \infty \) , \(q_{1}, q_{2}>0\) , and \(v_{n} = 8 K_{n} / 9\) .

 \(K_{n}/n \rightarrow 1\)  and \(n-K_{n} \rightarrow \infty \) , \(F\)  is \(\Delta \) -analytic and there exist \(\alpha , c>0\)  such that \(F(\rho -z) \sim c/z^{\alpha }\)  as \(z \rightarrow 0\)  with \(\mathrm {Re}(z)>0\) , and finally \(v_{n} = \frac{4(1+\alpha )n^{2}}{9\alpha (n-K_{n})}\) .
As for random paths, in the first regime, since \(F^{(2)}(b) < \infty \) , then essentially the degree of a typical face in \(M_{n,K_n}\)  has uniformly bounded variance and the Brownian map is naturally expected in the limit.
In the second regime, we force the number of vertices to be small compared to the number of edges. Clearly, this requires \(k_{0}=1\) , i.e. \(q_{1} > 0\) : in order to have much less vertices than edges, the map must have (many!) double edges, i.e. faces with degree 2. On the other hand we also need the extra assumption \(q_{2} > 0\)  to avoid periodicity issues which would require a more technical analysis. However, there is no additional regularity assumption on \(F\) .
Finally, in the last regime, when the number of vertices is asymptotically equivalent to the number of edges, we recall that \(F\)  is \( \Delta \) -analytic if there exist \(R>\rho \)  and \(\phi \in (0,\pi /2)\)  such that \(F\)  is analytic on \( \lbrace z : |z|<R, z \ne \rho , |\arg (z-1)|>\phi \rbrace \) , see [42]}. This condition on \(F\)  is crucial and Theorem REF  below shows a very different behaviour.
Our final result concerning random maps focuses on the case \(K_{n}/n \rightarrow A(\rho )\)  when the latter is finite. Here, we assume that the probability distribution with generating function \(F(\rho \,\cdot )/F(\rho )\)  is aperiodic and belongs to the domain of attraction of a stable law with index \(\alpha \in (1, 2]\) , i.e. that there exist \(\alpha \in (1,2]\) , \(\rho >0\) , \(m>0\) , and a slowly varying function \(L\)  such that for every \(0 \leqslant s < \rho \) ,
\(F(s)=F(\rho ) \mathopen {}\mathclose {\left(1-m+\frac{m s}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{\alpha } L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\right)}.\) 
In the case \(F^{(2)}(\rho ) < \infty \) , i.e. when \(\alpha =2\)  and \(L\)  admits a finite limit at infinity, say \(\ell \) , the law with generating function \(F(\rho \,\cdot )/F(\rho )\)  has a finite variance, equal to
\(\sigma ^{2} = 2\ell + m - m^{2}= \frac{\rho ^{2} F^{(2)}(\rho )}{F(\rho )} + \frac{\rho F^{\prime }(\rho )}{F(\rho )} - \mathopen {}\mathclose {\left(\frac{\rho F^{\prime }(\rho )}{F(\rho )}\right)}^{2}.\) 
In this case we set \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  for every \(n\geqslant 1\) , otherwise, we let \((r_{n})_{n \geqslant 1}\)  be a sequence such that \(r_{n}^{\alpha } \sim n L(r_{n})\)  as \(n \rightarrow \infty \) .
Example 6.3 Let us give an explicit example taken from [43]}. The weights
\(q_k = \frac{-\sqrt{\pi }}{2\Gamma (1-\alpha )} \mathopen {}\mathclose {\left(\frac{1}{4\alpha }\right)}^{k-1} \frac{\Gamma (-\alpha +k)}{\Gamma (\frac{1}{2}+k)}\qquad \text{for}k \geqslant 2.\) 
give rise to the generation function \(F\)  satisfying \(F(\alpha z)/F(\alpha ) = z + \alpha ^{-1} (1-z)^\alpha \) . In this case, \(\rho =\alpha \) , \(m=1\) , \(F(\alpha )=\alpha \) , \(F^{\prime }(\rho )=1\) , \(F^{(2)}(\rho )=1/2\)  when \(\alpha =2\) , \(L\)  is constant equal to 1, and finally \(A(\alpha ) = 1/\alpha \) . In particular, one can take \(r_{n}=n^{1/\alpha }\) .

The next statements are our last applications to random maps.
Theorem 6.4 
Under the previous assumptions, assume that \(K_{n}\)  takes the form
\(K_n = A(\rho ) n + \lambda _{n},\qquad \text{where}\qquad \frac{\lambda _{n}}{r_{n}} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }\lambda \in [-\infty ,\infty ].\) 
The following convergences hold for the Gromov–Hausdorff–Prokhorov topology:

If \(\lambda \in \mathbb {R}\)  and \(\alpha = 2\) , and
\(v_{n} = \frac{\rho F^{(2)}(\rho )}{F^{\prime }(\rho )} \frac{4n}{9} \mathbb {1}_{\lbrace F^{(2)}(\rho ) < \infty \rbrace }+ \frac{F(\rho )}{\rho F^{\prime }(\rho )}\frac{8 r_{n}^{2}}{9} \mathbb {1}_{\lbrace F^{(2)}(\rho ) = \infty \rbrace }.\) 
then
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), v_{n}^{-1/4} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }({M}, {D}, {m}),\) 
where \(({M}, {D}, {m})\)  is the Brownian map.

 If \(\lambda = -\infty \)  and \(\lambda _{n}/n \rightarrow 0\) , let \(\varepsilon _{n} \rightarrow 0\)  be defined by \(A(\rho (1-\varepsilon _{n})) = K_{n}/n\) , then for
\(v_{n} = \frac{\rho F^{(2)}(\rho )}{F^{\prime }(\rho )} \frac{4n}{9} \mathbb {1}_{\lbrace F^{(2)}(\rho ) < \infty \rbrace }+ \frac{4}{9} \frac{\alpha -1}{1-A(\rho )} \frac{|\lambda _{n}|}{\varepsilon _{n}} \mathbb {1}_{\lbrace F^{(2)}(\rho ) = \infty \rbrace },\) 
we have
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), v_{n}^{-1/4} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }({M}, {D}, {m}),\) 
where \(({M}, {D}, {m})\)  is the Brownian map.

 If \(\lambda = \infty \) , we further assume that \(\binom{2k-1}{k-1} q_{k} \rho ^{-k}\)  is regularly varying at infinity with some index \(-\beta < -2\) .
If \(F^{(2)}(\rho )<\infty \)  (which implies \(\beta \geqslant 3\) ), we assume furthermore that there exists \(c>(\beta -3)/\sigma ^{2}\)  such that \(\lambda _{n} \geqslant \sqrt{c \ln (n)}\) .
Then
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), (2\lambda _{n})^{-1/2} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)} \mathop {\longrightarrow }^{(d)}_{n \rightarrow \infty }({T}, {d}, {p})\) 
where \(({T}, {d}, {p})\)  is Aldous' Brownian tree coded by the standard Brownian excursion.
We shall establish these convergences by combining Theorem REF  with [35]}. Our last result concerns the first regime, \(\lambda \in \mathbb {R}\) , when \(\alpha \in (1,2)\) . In this case, we expect new limit spaces to appear, and in this direction here we establish a tightness result as a first step.
Theorem 6.5 
In the same framework as the previous theorem, still assuming that \(K_{n}\)  takes the form
\(K_n = A(\rho ) n + \lambda _{n},\qquad \text{where}\qquad \frac{\lambda _{n}}{r_{n}} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }\lambda \in \mathbb {R}.\) 
When \(\alpha \in (1,2)\) , the sequence
\(\mathopen {}\mathclose {\left(V(M_{n,K_n}), \frac{1}{\sqrt{r_{n}}} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)}_{n \geqslant 1}\) 
is tight for the Gromov–Hausdorff–Prokhorov topology and nondegenerate (in the sense that it does not converge towards a deterministic limit).

In the case of maps solely conditioned by their number of vertices, Le Gall & Miermont [25]} proved such a tightness result by showing that the label process
(which records the label of the vertices of the tree in depth-first search order) converges in distribution. This enabled them to show that all subsequential limits have the same Hausdorff dimension \(2\alpha \)  almost surely; furthermore some limit theorems then do not require the extraction of a subsequence, such as the convergence of the profile of distances to an independent uniform random vertex.
Here, Theorem REF  follows by a direct application of [35]} which only shows tightness of the label process, which does not provide any extra information on the limit spaces other than the scale of the typical distances.
In the companion paper [47]}, we extend Le Gall & Miermont's results which involved the so-called `continuous distance process' based on the excursion of stable Lévy process to more general Lévy processes with no negative jump.
Applied to the present setting of biconditioned maps, we obtain that all subsequential limits in Theorem REF  have the same Hausdorff, packing, and Minkowski dimensions, all equal to \(2\alpha \)  almost surely, independently of \(\lambda \in \mathbb {R}\) .
Finally, in the case \(\lambda =0\) , the limit spaces are the same as in [25]}.
Remark 6.6 It is interesting to note that in the case \(m<1\) , under an additional regularity assumption, the scaling limit of \({q}\) -Boltzmann planar maps conditioned on having a large number of edges is the Brownian CRT (see [31]} and [35]}), but an appropriate biconditioning also allows to escape this universality class.

[Proof of Theorems REF , REF , and REF ]
The paper [35]} specifically studies a model of uniform random maps `with prescribed degrees' in the sense that given, say, \(N\)  positive integers \(d_{N,1} \geqslant \cdots \geqslant d_{N,N}\) , a (rooted planar) map is sampled uniformly at random amongst all those with \(N\)  faces, with degrees \(2d_{N,1}, \dots , 2d_{N,N}\) . Then one wants to understand to behaviour of such a map \(M_{N}\)  as \(N \rightarrow \infty \)  in terms of the triangular array \((d_{N,i})_{N \geqslant i \geqslant 1}\) . It is shown in [35]} that
the sequence
\(\mathopen {}\mathclose {\left(V(M_{N}), \sigma _{N}^{-1/2} d_{\mathrm {gr}}, p_{\mathrm {unif}}\right)}_{n \geqslant 1}\qquad \text{where}\qquad \sigma _{N}^{2} = \sum _{i=1}^{N} d_{N,i} (d_{N,i}-1),\) 
is always tight for the Gromov–Hausdorff–Prokhorov topology.
Moreover, by Theorem 2 and Theorem 3 respectively in [35]}, as \(N \rightarrow \infty \) ,

if \(\sigma _{N}^{-1} d_{N,1} \rightarrow 0\) , then it converges in distribution towards \(\sqrt{2/3}\)  times the Brownian map \(({M}, {D}, {m})\) ;

if \(\sigma _{N}^{-1} d_{N,1} \rightarrow 1\) ,
then it converges in distribution towards \(\sqrt{2}\)  times the Brownian tree \(({T}, {d}, {p})\) .

Since two maps with same face degrees have the same \(w^{{q}}\) -weight, a size-conditioned Boltzmann map \(M_{n,K_n}\)  sampled from \(\mathbb {P}^{{q}}_{n,K_{n}}\)  can be seen as a mixture of this model with \(N=n-K_{n}\) : One first samples the degrees \(d_{n-K_{n},1} \geqslant \cdots \geqslant d_{n-K_{n},n-K_{n}}\)  randomly with an appropriate distribution, and then conditionally given this sequence, the map \(M_{n,K_n}\)  has the uniform distribution on maps with \(n-K_{n}\)  faces, with degrees respectively \(2d_{n-K_{n},1}, \dots , 2d_{n-K_{n},n-K_{n}}\) .
We then deduce that, once rescaled by a, now random, factor, the sequence of maps \((\sigma _{n-K_{n}}^{-1/2} M_{n,K_n})_{n \geqslant 1}\)  is tight, and it suffices to prove the convergence in probability \(d_{n-K_{n},1} / \sigma _{n-K_{n}} \rightarrow 0\)  as \(n \rightarrow \infty \)  to deduce that they converge to \(\sqrt{2/3}\)  times the Brownian map, or that \(d_{n-K_{n},1} / \sigma _{n-K_{n}} \rightarrow 1\)  in probability to deduce that they converge to \(\sqrt{2}\)  times the Brownian tree.
According to Sec. REF , the half face-degrees of \(M_{n,K_{n}}\)  are given by the nonnegative increments plus one of the Łukasiewicz path of the associated tree. Further, from Sec. REF , the latter has the law \(\mathbb {P}^{{\theta },+}_{n,K_{n}}\) , where \({\theta }\)  is defined by (REF ). If we denote by \(X^{n}_{k}\)  the increments of this path, then the quantity \(\sigma _{n-K_{n}}^{2}\)  has therefore the same law as
\(\sum _{k=1}^{n} X^{n}_{k} (X^{n}_{k}+1) = -1 + \sum _{k=1}^{n} (X^{n}_{k})^{2}.\) 
We infer from Theorems REF  and REF , with also Remark REF  in the finite variance regimes that:

The random factor \(\sigma _{n-K_{n}}\)  can be replaced by the deterministic factor used in Theorems REF , Theorem REF , and REF  respectively;

The ratio \(d_{n-K_{n},1} / \sigma _{n-K_{n}}\)  converges in probability to 0 as \(n \rightarrow \infty \)  in Theorem REF  as well as in Theorem REF  and in Theorem REF .

The ratio \(d_{n-K_{n},1} / \sigma _{n-K_{n}}\)  converges in probability to 1 as \(n \rightarrow \infty \)  in Theorem REF .

Our claim then follows from [35]}.
On the sum of the increments squared
In this section, we provide the proof of the convergence of the sum of the squares of the increments of the simply generated random path \(S^{n}\)  conditioned to end at \(S^{n}_{n} = x_{n}\) , as stated in
Theorems REF , REF ,
and finally REF  in the case \(\alpha =2\) . These results were needed when dealing with random maps just above.
Throughout this section, we shall denote by \(X^{n}_{k} = S^{n}_{k}-S^{n}_{k-1}\)  for every \(1 \leqslant k \leqslant n\)  the increments of the random bridge \(S^{n}\) .
Each regime requires a different argument and is studied in a separate subsection.
Recall the notation \(G\)  and \(\Psi \)  from (REF ).
In each regime studied from Sec. REF  to Sec. REF , we have \(x_{n}/n < \Psi (\rho )\)  for every \(n\)  large enough so we can define as in the proof of Proposition REF  a random variable \(\widehat{X}^{n}\)  with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) , where \(b_{n}\)  is such that \(\mathbb {E}[\widehat{X}^{n}] = \Psi (b_{n}) = x_{n}/n\) . We will use the explicit expressions of the first moments of \(\widehat{X}^{n}\) , which are given by (REF ).
We let further \((\widehat{S}^{n}_{i} ; i \geqslant 0)\)  denote a random walk with such i.i.d. steps.
On the other hand, in Sec. REF , we have \(x_{n}/n \geqslant \Psi (\rho )\) , so the latter is finite and we now define \(\widehat{X}\)  similarly but with \(b = \rho \)  instead of \(b_{n}\) ; then \((\widehat{S}_{i} ; i \geqslant 0)\)  shall denote a random walk with such i.i.d. steps.
In both cases, Lemma REF  shows that the paths \((\widehat{S}^{n}_{i} ; 0 \leqslant i \leqslant n)\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}^{n}_{n} = x_{n})\)  and \((\widehat{S}_{i} ; 0 \leqslant i \leqslant n)\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}_{n} = x_{n})\)  have the same law as our original path \(S^{n}\) .
The bulk regime
In this first subsection, we focus on Theorem REF , i.e. we assume that \(x_{n}/n\)  converges to some limit \(\gamma \in (i_{{p}}, \Psi (\rho ))\) .
Let \(b_{n}\)  and \(\widehat{X}^{n}\)  as above, then \(b_{n} \rightarrow b\)  where \(b \in (0, \rho )\)  is such that \(\Psi (b) = \gamma \) ; we let similarly \(\widehat{X}^{b}\)  have the generating function \(\widehat{G}^{b}(s) = G(b s) / G(b)\) .
The next result completes Theorem REF  and was used in the proof of Theorem REF .
Proposition A.1 
The following convergence holds in probability:
\(\frac{1}{n} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\mathbb {E}\big [(\widehat{X}^{b})^{2}\big ] = \frac{b^{2} G^{(2)}(b) + b G^{\prime }(b)}{G(b)}.\) 


According to Lemma REF , the law of the path \(S^{n}\)  and the random walk \(\widehat{S}^{n}\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}^{n}_{n}=x_{n})\)  are the same so we may replace the increments of the former by those of the latter. Let us therefore consider a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) , and recall their second moment from (REF ), namely
\(s^{2}_{n} = \mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ]= \frac{b_{n}^{2} G^{(2)}(b_{n}) + b_{n} G^{\prime }(b_{n})}{G(b_{n})}.\) 
Then \(s^{2}_{n} \rightarrow s^{2} = \mathbb {E}[(\widehat{X}^{b})^{2}]\) .
Fix \(\varepsilon >0\) ; the proof by truncation of the weak law of large numbers provided e.g. in [55]}, shows the following: let \(\Sigma _{n} = \sum _{k=1}^{n} ((\widehat{X}^{n}_{k})^{2} - s_{n}^{2})\)  and \(\Sigma _{n}^{\prime } = \sum _{k=1}^{n} ((\widehat{X}^{n}_{k})^{2} - s_{n}^{2}) \mathbb {1}_{\lbrace |(\widehat{X}^{n}_{k})^{2} - s_{n}^{2}| \leqslant n \varepsilon ^{3}\rbrace }\) , then

P(|n-E[n']| > n )
  E[|(Xn1)2 - sn2|] + n  P(|(Xn1)2 - sn2| > n 3)
2sn2 + -3  E[|(Xn1)2 - sn2| 1{|(Xn1)2 - sn2| > n 3}],

and
\(n^{-1} \mathopen {}\mathclose {\left|\mathbb {E}\mathopen {}\mathclose {\left[\Sigma _{n}^{\prime }\right]}\right|} \leqslant \mathbb {E}\mathopen {}\mathclose {\left[\mathopen {}\mathclose {\left|(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}\right|} \mathbb {1}_{\lbrace |(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}| > n \varepsilon ^{3}\rbrace }\right]}.\) 
Recall the fourth moment of \(\widehat{X}^{n}_{1}\)  from (REF ); since \(b_{n} \rightarrow b \in (0,\rho )\) , then \(\mathbb {E}[(\widehat{X}^{n}_{1})^{4}]\)  converges to the similar quantity with \(b\)  in place of \(b_{n}\)  (which equals \(\mathbb {E}[(\widehat{X}^{b})^{4}]\) ). Since \(s_{n}^{2} \rightarrow s^{2}\) , we deduce that \(((\widehat{X}^{n}_{1})^{2} - s_{n}^{2})_{n}\)  is uniformly bounded in \(L^{2}\)  and therefore the right-hand side in the preceding display converges to 0. Hence,
\(\lim _{n\rightarrow \infty } n^{-1} \mathbb {E}\mathopen {}\mathclose {\left[\Sigma _{n}^{\prime }\right]} = 0\qquad \text{and}\qquad \limsup _{n\rightarrow \infty } \mathbb {P}\mathopen {}\mathclose {\left(\mathopen {}\mathclose {\left|\Sigma _{n}-\mathbb {E}\mathopen {}\mathclose {\left[\Sigma _{n}^{\prime }\right]}\right|} > n \varepsilon \right)}\leqslant 2\varepsilon s^{2}.\) 
Since this holds for every \(\varepsilon >0\) , we conclude that \(n^{-1} \Sigma _{n}\)  converges to 0 in probability, so
\(\frac{1}{n} \sum _{k=1}^{n} (\widehat{X}_{k}^{n})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }s^{2},\) 
under the unconditional law. Then as in the proof of Proposition REF , by absolute continuity, we can transfer this convergence to the first half of the bridge, i.e. under \(\mathbb {P}( \, \cdot \mid \widehat{S}^{n}_{n}=x_{n})\) ; by time-reversal the same holds also for the second half of the bridge. Since this bridge of \(\widehat{S}^{n}\)  has the same law as our path \(S^{n}\) , this shows our assertion.

The small endpoint regime
We focus in this subsection on Theorem REF : we only assume that \(p(0), p(1) > 0\)  and we suppose that \(x_{n}/n \rightarrow 0\) .
The next result completes Theorem REF  and was used in the proof of Theorem REF .
Proposition A.2 
The following convergence in probability holds:
\(\frac{1}{x_{n}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }1.\) 


Exactly as in the proof of Proposition REF ,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}^{n}\) , and then conclude by absolute continuity and time-reversal. We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) , where \(b_{n} \in (0, \rho )\)  is such that \(\Psi (b_{n}) = x_{n} / n\) , so \(b_{n} \rightarrow 0\) .
We then derive from (REF ) the mean and variance of \((\widehat{X}^{n}_{1})^{2}\) , and further, by (REF ),
\(\mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ] = \frac{b_{n}^{2} G^{(2)}(b_{n}) + b_{n} G^{\prime }(b_{n})}{G(b_{n})}= \frac{x_{n}}{n} \mathopen {}\mathclose {\left(\frac{b_{n}^{2} G^{(2)}(b_{n})}{b_{n} G^{\prime }(b_{n})} + 1\right)}\mathop {\sim }^{}_{n \rightarrow \infty }\frac{x_{n}}{n},\) 
and similarly
\(\mathbb {E}\big [(\widehat{X}^{n}_{1})^{4}\big ]= \frac{x_{n}}{n} \mathopen {}\mathclose {\left(\frac{b_{n}^{4} G^{(4)}(b_{n}) + 6 b_{n}^{3} G^{(3)}(b_{n}) - 11 b_{n}^{2} G^{(2)}(b_{n})}{b_{n} G^{\prime }(b_{n})} + 6\right)},\) 
so
\(\mathrm {Var}\big ((\widehat{X}^{n}_{1})^{2}\big ) \mathop {\sim }^{}_{n \rightarrow \infty }6 \frac{x_{n}}{n}.\) 
Since \(x_{n} \rightarrow \infty \) , we infer that \(\frac{1}{x_{n}} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  converges to 1 in \(L^{2}\)  under the unconditioned probability, which suffices to conclude.

The large endpoint regime
In this subsection, we focus on Theorem REF : We now assume that \(x_{n}/n \rightarrow \infty \) , that \(G\)  is \(\Delta \) -analytic, and that there exist \(c, \rho , \alpha >0\)  such that \(G(\rho -z) \sim c z^{-\alpha }\)  as \(z \rightarrow 0\)  with \(\mathrm {Re}(z)>0\) .
The next result completes Theorem REF  and was used in the proof of Theorem REF .
Proposition A.3 
The following convergence in probability holds:
\(\frac{n}{x_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\frac{\alpha +1}{\alpha }.\) 


The proof is similar to that of Proposition REF . First,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}^{n}\) , and then conclude by absolute continuity and time-reversal that the same holds for the bridge, which has the same law as \(S^{n}\) . We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) .
Second, we rely on the moments of \((\widehat{X}^{n}_{1})^{2}\)  calculated in (REF ).
Indeed, we already noticed in (REF ) that, as \(n\rightarrow \infty \) ,
\(\mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ] \sim \frac{\alpha +1}{\alpha } \frac{x_{n}^{2}}{n^{2}}.\) 
Similarly, combining the expression of the moments from (REF ) and the derivatives of \(G\)  from (REF ) we deduce that as \(n\rightarrow \infty \) ,
\(\mathrm {Var}\big ((\widehat{X}^{n}_{1})^{2}\big )= O\mathopen {}\mathclose {\left(\frac{x_{n}^{4}}{n^{4}}\right)}.\) 
Therefore \(\frac{n}{x_{n}^{2}} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  converges to \((\alpha +1)/\alpha \)  in \(L^{2}\)  under the unconditioned probability, which suffices to conclude.

Stable regime with a large negative deviation
We treat in this subsection the regime described in Theorem REF , where the limit process is again a Brownian bridge.
Recall the assumptions of this theorem: we assume throughout that \(G^{\prime }(\rho ) < \infty \)  and that there exist \(\alpha \in (1,2]\)  and a slowly varying function \(L\)  such that for every \(0 \leqslant s < \rho \) ,
\(G(s)=G(\rho ) \mathopen {}\mathclose {\left(1-m+\frac{m s}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{\alpha } L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\right)}.\) 
Recall that in the case \(G^{(2)}(\rho ) < \infty \) , i.e. when \(\alpha =2\)  and \(L\)  admits a finite limit at infinity, say \(\ell \) , the law with generating function \(G(\rho \,\cdot )/G(\rho )\)  has a finite variance, equal to
\(\sigma ^{2} = 2\ell + m - m^{2}.\) 
In this case we set \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  for every \(n\geqslant 1\) , otherwise, we let \((r_{n})_{n \geqslant 1}\)  be a sequence such that \(r_{n}^{\alpha } \sim n L(r_{n})\)  as \(n \rightarrow \infty \) .
Finally, we let
\(x_n = m n + \lambda _{n},\qquad \text{where}\qquad \lambda _{n}/r_{n} \rightarrow -\infty \qquad \text{and}\qquad \lambda _{n}/n \rightarrow 0,\) 
and we write \(b_{n} = \rho (1-\varepsilon _{n})\) , which satisfies \(\Psi (b_{n}) = x_{n}/n\) , so \(\varepsilon _{n} \rightarrow 0\) .
Again, we prove a law of large number for the square of the increments; this completes Theorem REF  and was used in the proof of Theorem REF  in the case \(\lambda = -\infty \) .
Proposition A.4 
The following convergence in probability holds:
\(\frac{\varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\alpha -1 + \frac{m^{2}}{\sigma ^{2}} \mathbb {1}_{\lbrace G^{(2)}(\rho )<\infty \rbrace }.\) 


Again, we proceed exactly as in the proof of Proposition REF : First,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}^{n}\) , and then conclude by absolute continuity and time-reversal that the same holds for the bridge, which has the same law as \(S^{n}\) . We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}^{n}_{k})_{k \geqslant 1}\)  of i.i.d. copies of a random variable \(\widehat{X}^{n}\)  with generating function \(\widehat{G}^{n} = G(b_{n} \,\cdot ) / G(b_{n})\) .
Second, we rely on the moments of \((\widehat{X}^{n})^{2}\)  calculated in (REF ).
Note that the generating function \(\widehat{G}^{n}\)  satisfies the same assumption that \(G\)  in Sec. REF  and \(\widehat{X}^{n}\)  has the law of \(\xi ^{(b_{n})}\)  there.
Let us first treat the case where either \(\alpha < 2\)  or \(L\)  converges to \(\infty \) . In this case, combining (REF ) and (REF ) we obtain
\(\frac{n \varepsilon _{n}}{|\lambda _{n}|} \mathbb {E}\big [(\widehat{X}^{n}_{1})^{2}\big ]\mathop {\longrightarrow }^{}_{n \rightarrow \infty }\alpha -1.\) 
By the same argument, relying on the expression of the moments from (REF ) and the derivatives of \(G\)  from (REF ) we infer that as \(n\rightarrow \infty \) ,
\(\frac{n \varepsilon _{n}}{|\lambda _{n}|} \mathbb {E}\big [(\widehat{X}^{n}_{1})^{4}\big ]\sim \frac{n \varepsilon _{n}}{|\lambda _{n}|} \frac{b_{n}^{4} G^{(4)}(b_{n})}{G(b_{n})}\sim \frac{(\alpha -1)(\alpha -2)(\alpha -3)}{\varepsilon _{n}^{2}}.\) 
Recall also that we deduced just below (REF ) that \(r_{n} \varepsilon _{n} \rightarrow \infty \) , in addition to \(|\lambda _{n}| \rightarrow \infty \) , hence
\(n \mathopen {}\mathclose {\left(\frac{\varepsilon _{n}}{|\lambda _{n}|}\right)}^{2} \mathbb {E}\big [(\widehat{X}^{n}_{1})^{4}\big ] \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
We conclude in this case that \(\frac{\varepsilon _{n}}{|\lambda _{n}|} \sum _{k=1}^{n} (\widehat{X}^{n}_{k})^{2}\)  converges to \(\alpha -1\)  in \(L^{2}\)  under the unconditioned probability, which suffices for our claim.
It remains to treat the case where \(\alpha =2\)  and \(L\)  admits a finite and positive limit, say \(\ell \) . Here we use (REF ) instead.
Moreover in this case \(G^{(2)}(\rho ) < \infty \) ; let \(\widehat{X}\)  have the law with generating function \(G(\rho \,\cdot )/G(\rho )\) , so \(\mathbb {E}[\widehat{X}] = m\)  and let \(s^{2} = \mathbb {E}[(\widehat{X})^{2}]\) , which is given by (REF ), and notice that since \(2\ell = \mathbb {E}[\widehat{X}(\widehat{X}-1)]\) , then \(s^{2} = 2\ell + m\) .
Similarly let \(s^{2}_{n} = \mathbb {E}[(\widehat{X}^{n}_{1})^{2}]\) , then by continuity \(s^{2}_{n} \rightarrow s^{2}\) .
The claim can be proved very much as in the proof of Proposition REF . Recall from there that all we need to prove is that for any \(\varepsilon > 0\) ,
\(\mathbb {E}\Big [\big |(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}\big | \mathbb {1}_{\lbrace |(\widehat{X}^{n}_{1})^{2} - s_{n}^{2}| > n \varepsilon ^{3}\rbrace }\Big ] \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
However here we cannot rely on a uniform bound for the fourth moment, and we argue instead by absolute continuity.
Indeed, for every \(k \geqslant 0\) , we have
\(\mathbb {P}(\widehat{X} = k) = \frac{\rho ^{k} p(k)}{G(\rho )}\qquad \text{and}\qquad \mathbb {P}(\widehat{X}^{n}_{1} = k) = \frac{b_{n}^{k} p(k)}{G(b_{n})}.\) 
Consequently, since \(b_{n} \leqslant \rho \)  for every \(n\)  and \(G(b_{n}) \rightarrow G(\rho )\) , then

E[|(Xn1)2 - sn2| 1{|(Xn1)2 - sn2| > n 3}]
= G()G(bn) E[|(X)2 - sn2| 1{|(X)2 - sn2| > n 3} (bn)k]
G()G(bn) E[|(X)2 - sn2| 1{|(X)2 - sn2| > n 3}],

which tends to 0. We conclude as in the proof of Proposition REF  that \(\frac{1}{n} \sum _{k=1}^{n} (\widehat{X}^{n}_{k})^{2}\)  converges to \(s^{2}\)  in probability.
Our claim then follows from (REF ) together with the fact that the constant there equals \(s^{2} - m^{2}\) .
Notice that when \(L\)  has a finite limit at infinity, say \(\ell \) , then \(|\lambda _{n}| / \varepsilon _{n} \sim (2\ell + m - m^{2}) n = (s^{2} - m^{2}) n\)  by (REF )
so \(n^{-1} \sum _{k=1}^{n} (X^{n}_{k})^{2}\)  converges in probability to \(s^{2} = \mathbb {E}[\widehat{X}^{2}]\) , which is consistent with Proposition REF .

Gaussian regime with a small deviation
We finally treat in this subsection the regime described in Theorem REF , in the case \(\alpha =2\) , and one last time, we prove a law of large number for the square of the increments; this was used in the proof of Theorem REF  in the case \(\alpha =2\) .
We work under the same assumptions on \(G\)  as in the previous subsection with \(\alpha =2\) , that is
\(G(s)=G(\rho ) \bigg (1-m+\frac{m s}{\rho }+\mathopen {}\mathclose {\left(1-\frac{s}{\rho }\right)}^{2} L\mathopen {}\mathclose {\left(\frac{1}{1-s/\rho }\right)}\bigg ),\) 
with \(r_{n}\)  defined as in Sec. REF .
However here we assume that there exists \(\lambda \in \mathbb {R}\)  such that
\(x_n = m n + \lambda _{n},\qquad \text{where}\qquad \lambda _{n}/r_{n} \rightarrow \lambda .\) 
As opposed to the preceding regimes, we shall use a fixed distribution for the equivalent random walk. Precisely, we let \(\widehat{X}\)  have the law with generating function \(\widehat{G} = G(\rho \,\cdot )/G(\rho )\) , so its moments are given by (REF ) with \(b = \rho \) . We let \(\widehat{S}\)  denote a random walk with such i.i.d. steps, so Lemma REF  shows that the path \((\widehat{S}_{i} ; 0 \leqslant i \geqslant n)\)  under \(\mathbb {P}(\,\cdot \mid \widehat{S}_{n} = x_{n})\)  has the same law as our original path \(S^{n}\) .
Proposition A.5 
When \(\alpha =2\) , the following convergence in probability holds:
\(r_{n}^{-2} \sum _{k=1}^{n} (X^{n}_{k})^{2} \mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }2 + \frac{2m^{2}}{\sigma ^{2}} \mathbb {1}_{\lbrace G^{(2)}(\rho ) < \infty \rbrace }.\) 


As now usual,
it is sufficient to work under the unconditional law of the random walk \(\widehat{S}\) , and then conclude by absolute continuity and time-reversal that the same holds for the bridge, which has the same law as \(S^{n}\) . We therefore replace the \(X^{n}_{k}\) 's by a sequence \((\widehat{X}_{k})_{k \geqslant 1}\)  of i.i.d. random variables with generating function \(\widehat{G}\) .
The latter belong to the domain of attraction of a Gaussian distribution;
in the case where \(L\)  has a finite limit, we have \(\widehat{G}^{(2)}(\rho ) < \infty \)  so the \((\widehat{X}_{k})^{2}\) 's have finite mean and the claim simply follows from the law of large numbers combined with the fact that \(r_{n} = \sqrt{n \sigma ^{2}/2}\)  :
\(n^{-1} \sum _{k=1}^{n} (X^{n}_{k})^{2}\mathop {\longrightarrow }^{\mathbb {P}}_{n \rightarrow \infty }\mathbb {E}[\widehat{X}^{2}]=\sigma ^{2}+m^{2}\) 
In particular, the convergence of Proposition REF  also holds for \(b=\rho \)  when \(G^{(2)}(\rho ) < \infty \) .
This can be extended to the case \(\widehat{G}^{(2)}(\rho ) =\infty \)  by looking at the first two truncated moments, as in e.g. [35]}. We reproduce the argument for the reader's convenience. Let \(L_{1}\)  be the slowly varying function defined by \(L_{1}(x) = \mathbb {E}[\widehat{X}_{1}^{2} \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant x\rbrace }]\)  for every \(x > 0\) .
As in e.g. [57]} we have \((L_{1}(x)-m) \sim 2 L(x)\)  as \(x \rightarrow \infty \) . Further, by [2]}, the ratio \(x^2 \mathbb {P}(\widehat{X}_{1} \geqslant x) / L_{1}(x)\)  converges to \((2-\alpha )/2 = 0\)  as \(x \rightarrow \infty \) .
Then for every \(\varepsilon > 0\) , since \(L_1\)  is slowly varying, then \(L_1(\varepsilon r_n) \sim L_1(r_n)\)  and so
\(\mathbb {P}\mathopen {}\mathclose {\left(r_n^{-1} \max _{1 \leqslant i \leqslant n} \widehat{X}_i \geqslant \varepsilon \right)}\leqslant n \mathbb {P}\mathopen {}\mathclose {\left(\overline{X}_{1} \geqslant \varepsilon r_n\right)}= o\mathopen {}\mathclose {\left(n (\varepsilon r_n)^{-2} L_{1}(\varepsilon r_n)\right)}= o\mathopen {}\mathclose {\left(n r_n^{-2} L_{1}(r_n)\right)}= o(1),\) 
where we used that \(n r_n^{-2} \sim L(r_n)^{-1}\)  and that the ratio \(L_{1}(r_n)/L(r_n)\)  is bounded.
Further,
\(\mathbb {E}\bigg [r_{n}^{-2} \sum _{1 \leqslant i \leqslant n} \widehat{X}_{i}^2 \mathbb {1}_{\lbrace |\widehat{X}_{i}| \leqslant \varepsilon r_{n}\rbrace }\bigg ]= n r_{n}^{-2} \mathbb {E}\mathopen {}\mathclose {\left[\widehat{X}_{1}^2 \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant \varepsilon r_{n}\rbrace }\right]}\sim n r_{n}^{-2} L_{1}(r_{n})\sim \frac{L_{1}(r_{n})}{L(r_{n})},\) 
and, similarly, the variance of this quantity equals
\(n r_{n}^{-4} \mathrm {Var}\mathopen {}\mathclose {\left(\widehat{X}_{1}^2 \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant \varepsilon r_{n}\rbrace }\right)}\leqslant \varepsilon ^2 n r_{n}^{-2} \mathbb {E}\mathopen {}\mathclose {\left[\widehat{X}_{1}^2 \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant \varepsilon r_{n}\rbrace }\right]}\sim \varepsilon ^2 \frac{L_{1}(r_{n})}{L(r_{n})}.\) 
Since we have previously shown that with high probability, \(|\widehat{X}_{i}| \leqslant \varepsilon r_n\)  for every \(i \leqslant n\) , then we conclude that \(r_n^{-2} \sum _{1 \leqslant i \leqslant n} \widehat{X}_{i}^2\)  converges in probability to the limit of the ratio \(L_{1}/L\) , which is 2. The completes the proof.
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